Topology - Northern Illinois Universityalinner/Math650/Topology.pdf · Topology Linn er Evidence of...
594
Topology Linn´ er Evidence of topology. The logarithm. Existence? An example involving the logarithm. Work and flux. Potential and curl. Potential and curl. Flux without divergence? Detecting differences. How do the domains differ? The point with another point. Adding a point to R 2 . If not from a metric, what is topology? Open intervals. Open sets in R Open sets in Topology Anders Linn´ er [email protected] Get in shape!
Transcript of Topology - Northern Illinois Universityalinner/Math650/Topology.pdf · Topology Linn er Evidence of...
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology
Anders [email protected]
Get in shape!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Starting Point
How is the logarithm defined?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Definition.
Let X be the collection of differentiable real-valued functionsdefined on the domain of positive numbers.
Let Y be the collection of real-valued functions defined on thedomain of positive numbers.
Let ξ : X → Y be the operator that is defined by ξ(x) ∈ Ysuch that ξ(x)(t) = x ′(t).
Consider the function y ∈ Y given by y(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Definition.
Let X be the collection of differentiable real-valued functionsdefined on the domain of positive numbers.
Let Y be the collection of real-valued functions defined on thedomain of positive numbers.
Let ξ : X → Y be the operator that is defined by ξ(x) ∈ Ysuch that ξ(x)(t) = x ′(t).
Consider the function y ∈ Y given by y(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Definition.
Let X be the collection of differentiable real-valued functionsdefined on the domain of positive numbers.
Let Y be the collection of real-valued functions defined on thedomain of positive numbers.
Let ξ : X → Y be the operator that is defined by ξ(x) ∈ Ysuch that ξ(x)(t) = x ′(t).
Consider the function y ∈ Y given by y(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Definition.
Let X be the collection of differentiable real-valued functionsdefined on the domain of positive numbers.
Let Y be the collection of real-valued functions defined on thedomain of positive numbers.
Let ξ : X → Y be the operator that is defined by ξ(x) ∈ Ysuch that ξ(x)(t) = x ′(t).
Consider the function y ∈ Y given by y(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Existence?
Is there any x ∈ X such that y = ξ(x)?
In the context of differential equations the question may bephrased as an initial value problem
dx
dt=
1
tsuch that x(1) = 0.
This separable first-order equation leads directly to the formula
x(t) =
∫ t
1
1
udu.
To understand this formula one must understand the theory ofintegration leading up to the fundamental theorem of calculus,which in this particular case asserts that the derivative of theexpression is in fact 1
t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Existence?
Is there any x ∈ X such that y = ξ(x)?
In the context of differential equations the question may bephrased as an initial value problem
dx
dt=
1
tsuch that x(1) = 0.
This separable first-order equation leads directly to the formula
x(t) =
∫ t
1
1
udu.
To understand this formula one must understand the theory ofintegration leading up to the fundamental theorem of calculus,which in this particular case asserts that the derivative of theexpression is in fact 1
t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Existence?
Is there any x ∈ X such that y = ξ(x)?
In the context of differential equations the question may bephrased as an initial value problem
dx
dt=
1
tsuch that x(1) = 0.
This separable first-order equation leads directly to the formula
x(t) =
∫ t
1
1
udu.
To understand this formula one must understand the theory ofintegration leading up to the fundamental theorem of calculus,which in this particular case asserts that the derivative of theexpression is in fact 1
t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Existence?
Is there any x ∈ X such that y = ξ(x)?
In the context of differential equations the question may bephrased as an initial value problem
dx
dt=
1
tsuch that x(1) = 0.
This separable first-order equation leads directly to the formula
x(t) =
∫ t
1
1
udu.
To understand this formula one must understand the theory ofintegration leading up to the fundamental theorem of calculus,which in this particular case asserts that the derivative of theexpression is in fact 1
t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Is it OK?
It is of course immediate that the value of the expression is 0when t = 1.
It is crucial here that t is assumed to satisfy t > 0. With thisassumption satisfied, and the fact that the function integratedis continuous on the domain of all positive numbers, means thebasic calculus theory of Riemann integration suffices here.
Once existence is established it is useful to introduce a name,and the most common choice is ln.
It follows that ln(1) = 0 and ln′(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Is it OK?
It is of course immediate that the value of the expression is 0when t = 1.
It is crucial here that t is assumed to satisfy t > 0. With thisassumption satisfied, and the fact that the function integratedis continuous on the domain of all positive numbers, means thebasic calculus theory of Riemann integration suffices here.
Once existence is established it is useful to introduce a name,and the most common choice is ln.
It follows that ln(1) = 0 and ln′(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Is it OK?
It is of course immediate that the value of the expression is 0when t = 1.
It is crucial here that t is assumed to satisfy t > 0. With thisassumption satisfied, and the fact that the function integratedis continuous on the domain of all positive numbers, means thebasic calculus theory of Riemann integration suffices here.
Once existence is established it is useful to introduce a name,and the most common choice is ln.
It follows that ln(1) = 0 and ln′(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Is it OK?
It is of course immediate that the value of the expression is 0when t = 1.
It is crucial here that t is assumed to satisfy t > 0. With thisassumption satisfied, and the fact that the function integratedis continuous on the domain of all positive numbers, means thebasic calculus theory of Riemann integration suffices here.
Once existence is established it is useful to introduce a name,and the most common choice is ln.
It follows that ln(1) = 0 and ln′(t) = 1t .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
An interesting scalar field.
Let R be the real numbers and write R2 for R× R.
Let D = R2 − {(0, 0)} and consider the scalar field f : D → Rgiven by
f (x , y) =1
2ln(x2 + y2).
This function is well-defined since x2 + y2 > 0 whenever(x , y) 6= (0, 0).
To further analyze it, take the two partial derivatives
∂f
∂x=
x
x2 + y2,
∂f
∂y=
y
x2 + y2.
This produces the so-called gradient vector field ∇f on D.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
An interesting scalar field.
Let R be the real numbers and write R2 for R× R.
Let D = R2 − {(0, 0)} and consider the scalar field f : D → Rgiven by
f (x , y) =1
2ln(x2 + y2).
This function is well-defined since x2 + y2 > 0 whenever(x , y) 6= (0, 0).
To further analyze it, take the two partial derivatives
∂f
∂x=
x
x2 + y2,
∂f
∂y=
y
x2 + y2.
This produces the so-called gradient vector field ∇f on D.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
An interesting scalar field.
Let R be the real numbers and write R2 for R× R.
Let D = R2 − {(0, 0)} and consider the scalar field f : D → Rgiven by
f (x , y) =1
2ln(x2 + y2).
This function is well-defined since x2 + y2 > 0 whenever(x , y) 6= (0, 0).
To further analyze it, take the two partial derivatives
∂f
∂x=
x
x2 + y2,
∂f
∂y=
y
x2 + y2.
This produces the so-called gradient vector field ∇f on D.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
An interesting scalar field.
Let R be the real numbers and write R2 for R× R.
Let D = R2 − {(0, 0)} and consider the scalar field f : D → Rgiven by
f (x , y) =1
2ln(x2 + y2).
This function is well-defined since x2 + y2 > 0 whenever(x , y) 6= (0, 0).
To further analyze it, take the two partial derivatives
∂f
∂x=
x
x2 + y2,
∂f
∂y=
y
x2 + y2.
This produces the so-called gradient vector field ∇f on D.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
An interesting scalar field.
Let R be the real numbers and write R2 for R× R.
Let D = R2 − {(0, 0)} and consider the scalar field f : D → Rgiven by
f (x , y) =1
2ln(x2 + y2).
This function is well-defined since x2 + y2 > 0 whenever(x , y) 6= (0, 0).
To further analyze it, take the two partial derivatives
∂f
∂x=
x
x2 + y2,
∂f
∂y=
y
x2 + y2.
This produces the so-called gradient vector field ∇f on D.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Higher derivatives.
Take one more partial derivative and get
∂2f
∂x2=
y2 − x2
(x2 + y2)2,
∂2f
∂x∂y=
∂2f
∂y∂x= − 2xy
(x2 + y2)2,
and∂2f
∂y2=
x2 − y2
(x2 + y2)2.
If one adds the first and the last of the three formulas it is seenthat f satisfies Laplace’s equation, ∆f = 0, so f is a so-calledharmonic function.
Another way to express this is via the divergence, so∇ · ∇f = 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Higher derivatives.
Take one more partial derivative and get
∂2f
∂x2=
y2 − x2
(x2 + y2)2,
∂2f
∂x∂y=
∂2f
∂y∂x= − 2xy
(x2 + y2)2,
and∂2f
∂y2=
x2 − y2
(x2 + y2)2.
If one adds the first and the last of the three formulas it is seenthat f satisfies Laplace’s equation, ∆f = 0, so f is a so-calledharmonic function.
Another way to express this is via the divergence, so∇ · ∇f = 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Higher derivatives.
Take one more partial derivative and get
∂2f
∂x2=
y2 − x2
(x2 + y2)2,
∂2f
∂x∂y=
∂2f
∂y∂x= − 2xy
(x2 + y2)2,
and∂2f
∂y2=
x2 − y2
(x2 + y2)2.
If one adds the first and the last of the three formulas it is seenthat f satisfies Laplace’s equation, ∆f = 0, so f is a so-calledharmonic function.
Another way to express this is via the divergence, so∇ · ∇f = 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Work and flux.
Let S1 = f −1(0) and parameterize it using γ : [0, 2π]→ S1
given by γ(t) = (cos(t), sin(t)).
This induces counter-clockwise orientation on S1.
Since on S1 the gradient vector field is given by ∇f = (x , y)and the unit tangent vector field by T = (−y , x), it followsthat ∇f · T = 0, and therefore no work is performed by thevector field along S1.
Let N = (x , y) be the ‘outward’ pointing unit normal vector.
It follows that ∇f · N = 1 and the flux outwards through S1 is2π, the total length once around S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Work and flux.
Let S1 = f −1(0) and parameterize it using γ : [0, 2π]→ S1
given by γ(t) = (cos(t), sin(t)).
This induces counter-clockwise orientation on S1.
Since on S1 the gradient vector field is given by ∇f = (x , y)and the unit tangent vector field by T = (−y , x), it followsthat ∇f · T = 0, and therefore no work is performed by thevector field along S1.
Let N = (x , y) be the ‘outward’ pointing unit normal vector.
It follows that ∇f · N = 1 and the flux outwards through S1 is2π, the total length once around S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Work and flux.
Let S1 = f −1(0) and parameterize it using γ : [0, 2π]→ S1
given by γ(t) = (cos(t), sin(t)).
This induces counter-clockwise orientation on S1.
Since on S1 the gradient vector field is given by ∇f = (x , y)and the unit tangent vector field by T = (−y , x), it followsthat ∇f · T = 0, and therefore no work is performed by thevector field along S1.
Let N = (x , y) be the ‘outward’ pointing unit normal vector.
It follows that ∇f · N = 1 and the flux outwards through S1 is2π, the total length once around S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Work and flux.
Let S1 = f −1(0) and parameterize it using γ : [0, 2π]→ S1
given by γ(t) = (cos(t), sin(t)).
This induces counter-clockwise orientation on S1.
Since on S1 the gradient vector field is given by ∇f = (x , y)and the unit tangent vector field by T = (−y , x), it followsthat ∇f · T = 0, and therefore no work is performed by thevector field along S1.
Let N = (x , y) be the ‘outward’ pointing unit normal vector.
It follows that ∇f · N = 1 and the flux outwards through S1 is2π, the total length once around S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Work and flux.
Let S1 = f −1(0) and parameterize it using γ : [0, 2π]→ S1
given by γ(t) = (cos(t), sin(t)).
This induces counter-clockwise orientation on S1.
Since on S1 the gradient vector field is given by ∇f = (x , y)and the unit tangent vector field by T = (−y , x), it followsthat ∇f · T = 0, and therefore no work is performed by thevector field along S1.
Let N = (x , y) be the ‘outward’ pointing unit normal vector.
It follows that ∇f · N = 1 and the flux outwards through S1 is2π, the total length once around S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Potential and curl.
Anytime a vector field V is given by V = ∇f , then V is said tohave the potential f .
In this case the work performed by V along a path P from p toq is given by ∫
P
∇f · T = [f ]qp = f (q)− f (p).
Observe that this does not depend on the whole path, only theendpoints.
Moreover, if the path is closed so that q = p, then the totalwork is zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Potential and curl.
Anytime a vector field V is given by V = ∇f , then V is said tohave the potential f .
In this case the work performed by V along a path P from p toq is given by ∫
P
∇f · T = [f ]qp = f (q)− f (p).
Observe that this does not depend on the whole path, only theendpoints.
Moreover, if the path is closed so that q = p, then the totalwork is zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Potential and curl.
Anytime a vector field V is given by V = ∇f , then V is said tohave the potential f .
In this case the work performed by V along a path P from p toq is given by ∫
P
∇f · T = [f ]qp = f (q)− f (p).
Observe that this does not depend on the whole path, only theendpoints.
Moreover, if the path is closed so that q = p, then the totalwork is zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Potential and curl.
Anytime a vector field V is given by V = ∇f , then V is said tohave the potential f .
In this case the work performed by V along a path P from p toq is given by ∫
P
∇f · T = [f ]qp = f (q)− f (p).
Observe that this does not depend on the whole path, only theendpoints.
Moreover, if the path is closed so that q = p, then the totalwork is zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In the example it would therefore be true that the total workaround S1 is zero, which is in agreement with the previousreasoning.
The ‘2-dimensional curl’, which for V = (P,Q) is given by
∂Q
∂x− ∂P
∂y,
must vanish for a smooth vector field with potential.
The 2-dimensional version of Stokes’ theorem,∫boundary of D
V · T =
∫∫D
(∇× (P,Q, 0))3,
will also produce zero work around S1 when the vector field hasa potential.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In the example it would therefore be true that the total workaround S1 is zero, which is in agreement with the previousreasoning.
The ‘2-dimensional curl’, which for V = (P,Q) is given by
∂Q
∂x− ∂P
∂y,
must vanish for a smooth vector field with potential.
The 2-dimensional version of Stokes’ theorem,∫boundary of D
V · T =
∫∫D
(∇× (P,Q, 0))3,
will also produce zero work around S1 when the vector field hasa potential.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In the example it would therefore be true that the total workaround S1 is zero, which is in agreement with the previousreasoning.
The ‘2-dimensional curl’, which for V = (P,Q) is given by
∂Q
∂x− ∂P
∂y,
must vanish for a smooth vector field with potential.
The 2-dimensional version of Stokes’ theorem,∫boundary of D
V · T =
∫∫D
(∇× (P,Q, 0))3,
will also produce zero work around S1 when the vector field hasa potential.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Recall that ∫boundary of D
V · N =
∫∫D
∇ · V .
As seen previously, the gradient vector field has outward flux2π through S1.
What is the source?
The divergence of the gradient vector field is zero everywhereon D!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Recall that ∫boundary of D
V · N =
∫∫D
∇ · V .
As seen previously, the gradient vector field has outward flux2π through S1.
What is the source?
The divergence of the gradient vector field is zero everywhereon D!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Recall that ∫boundary of D
V · N =
∫∫D
∇ · V .
As seen previously, the gradient vector field has outward flux2π through S1.
What is the source?
The divergence of the gradient vector field is zero everywhereon D!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Recall that ∫boundary of D
V · N =
∫∫D
∇ · V .
As seen previously, the gradient vector field has outward flux2π through S1.
What is the source?
The divergence of the gradient vector field is zero everywhereon D!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Consider the scalar field g : R2 → R given by
g(x , y) =1
2(x2 + y2 + 1).
Again the gradient vector field restricted to S1 is given by(x , y).
The work performed is always zero.
The outward flux through S1 is again 2π.
The curl of the gradient vanishes again.
So far there is no difference.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Consider the scalar field g : R2 → R given by
g(x , y) =1
2(x2 + y2 + 1).
Again the gradient vector field restricted to S1 is given by(x , y).
The work performed is always zero.
The outward flux through S1 is again 2π.
The curl of the gradient vanishes again.
So far there is no difference.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Consider the scalar field g : R2 → R given by
g(x , y) =1
2(x2 + y2 + 1).
Again the gradient vector field restricted to S1 is given by(x , y).
The work performed is always zero.
The outward flux through S1 is again 2π.
The curl of the gradient vanishes again.
So far there is no difference.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Consider the scalar field g : R2 → R given by
g(x , y) =1
2(x2 + y2 + 1).
Again the gradient vector field restricted to S1 is given by(x , y).
The work performed is always zero.
The outward flux through S1 is again 2π.
The curl of the gradient vanishes again.
So far there is no difference.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Consider the scalar field g : R2 → R given by
g(x , y) =1
2(x2 + y2 + 1).
Again the gradient vector field restricted to S1 is given by(x , y).
The work performed is always zero.
The outward flux through S1 is again 2π.
The curl of the gradient vanishes again.
So far there is no difference.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
Consider the scalar field g : R2 → R given by
g(x , y) =1
2(x2 + y2 + 1).
Again the gradient vector field restricted to S1 is given by(x , y).
The work performed is always zero.
The outward flux through S1 is again 2π.
The curl of the gradient vanishes again.
So far there is no difference.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
When the divergence is calculated one sees that it is constanteverywhere with value 2.
When this is integrated over the region enclosed by S1 one getsthe value 2π, which is the flux.
The vector fields along S1 are the same in the two cases.
The domains of the scalar fields f and g are different!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
When the divergence is calculated one sees that it is constanteverywhere with value 2.
When this is integrated over the region enclosed by S1 one getsthe value 2π, which is the flux.
The vector fields along S1 are the same in the two cases.
The domains of the scalar fields f and g are different!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
When the divergence is calculated one sees that it is constanteverywhere with value 2.
When this is integrated over the region enclosed by S1 one getsthe value 2π, which is the flux.
The vector fields along S1 are the same in the two cases.
The domains of the scalar fields f and g are different!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Flux without divergence?
When the divergence is calculated one sees that it is constanteverywhere with value 2.
When this is integrated over the region enclosed by S1 one getsthe value 2π, which is the flux.
The vector fields along S1 are the same in the two cases.
The domains of the scalar fields f and g are different!
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
If one believes that [0, 1) and (0, 1) are isomorphic as sets viaφ : [0, 1)→ (0, 1), then R2 and D are isomorphic as sets.
To see this, use the isomorphism φ and extend it to a mapfrom [0, 1] to (0, 1].
Now use the identity on R2 − [0, 1] to itself, and the extendedmap on the rest of the space.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
If one believes that [0, 1) and (0, 1) are isomorphic as sets viaφ : [0, 1)→ (0, 1), then R2 and D are isomorphic as sets.
To see this, use the isomorphism φ and extend it to a mapfrom [0, 1] to (0, 1].
Now use the identity on R2 − [0, 1] to itself, and the extendedmap on the rest of the space.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
If one believes that [0, 1) and (0, 1) are isomorphic as sets viaφ : [0, 1)→ (0, 1), then R2 and D are isomorphic as sets.
To see this, use the isomorphism φ and extend it to a mapfrom [0, 1] to (0, 1].
Now use the identity on R2 − [0, 1] to itself, and the extendedmap on the rest of the space.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
It is tempting to dismiss the question of whether the puncturedplane is topologically different than the plane by claiming thatthe punctured plane is not closed.
This is, however, not the issue here.
The two spaces are not being compared as subsets of theplane, but rather as topological spaces in their own right.
The punctured plane has the restriction of the standard metrictopology available in the plane.
They differ as metric spaces as the punctured plane is notcomplete.
The thing is that completeness is not a topological property.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
It is tempting to dismiss the question of whether the puncturedplane is topologically different than the plane by claiming thatthe punctured plane is not closed.
This is, however, not the issue here.
The two spaces are not being compared as subsets of theplane, but rather as topological spaces in their own right.
The punctured plane has the restriction of the standard metrictopology available in the plane.
They differ as metric spaces as the punctured plane is notcomplete.
The thing is that completeness is not a topological property.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
It is tempting to dismiss the question of whether the puncturedplane is topologically different than the plane by claiming thatthe punctured plane is not closed.
This is, however, not the issue here.
The two spaces are not being compared as subsets of theplane, but rather as topological spaces in their own right.
The punctured plane has the restriction of the standard metrictopology available in the plane.
They differ as metric spaces as the punctured plane is notcomplete.
The thing is that completeness is not a topological property.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
It is tempting to dismiss the question of whether the puncturedplane is topologically different than the plane by claiming thatthe punctured plane is not closed.
This is, however, not the issue here.
The two spaces are not being compared as subsets of theplane, but rather as topological spaces in their own right.
The punctured plane has the restriction of the standard metrictopology available in the plane.
They differ as metric spaces as the punctured plane is notcomplete.
The thing is that completeness is not a topological property.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
It is tempting to dismiss the question of whether the puncturedplane is topologically different than the plane by claiming thatthe punctured plane is not closed.
This is, however, not the issue here.
The two spaces are not being compared as subsets of theplane, but rather as topological spaces in their own right.
The punctured plane has the restriction of the standard metrictopology available in the plane.
They differ as metric spaces as the punctured plane is notcomplete.
The thing is that completeness is not a topological property.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
It is tempting to dismiss the question of whether the puncturedplane is topologically different than the plane by claiming thatthe punctured plane is not closed.
This is, however, not the issue here.
The two spaces are not being compared as subsets of theplane, but rather as topological spaces in their own right.
The punctured plane has the restriction of the standard metrictopology available in the plane.
They differ as metric spaces as the punctured plane is notcomplete.
The thing is that completeness is not a topological property.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
The critical hurdle is to figure out how to mathematicallycapture the presence of the ‘hole’.
This is where the fundamental group will become the key actor,and algebraic topology the general tool.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
How do the domains differ?
The critical hurdle is to figure out how to mathematicallycapture the presence of the ‘hole’.
This is where the fundamental group will become the key actor,and algebraic topology the general tool.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
As seen, adding a point to an infinite set may not change theview of the set.
Identify R2 with C and consider the map η : [0, 1)→ C givenby η(t) = e2πit .
Observe that η([0, 1)) = S1.
This map is in fact a set-isomorphism from [0, 1) to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
As seen, adding a point to an infinite set may not change theview of the set.
Identify R2 with C and consider the map η : [0, 1)→ C givenby η(t) = e2πit .
Observe that η([0, 1)) = S1.
This map is in fact a set-isomorphism from [0, 1) to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
As seen, adding a point to an infinite set may not change theview of the set.
Identify R2 with C and consider the map η : [0, 1)→ C givenby η(t) = e2πit .
Observe that η([0, 1)) = S1.
This map is in fact a set-isomorphism from [0, 1) to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
As seen, adding a point to an infinite set may not change theview of the set.
Identify R2 with C and consider the map η : [0, 1)→ C givenby η(t) = e2πit .
Observe that η([0, 1)) = S1.
This map is in fact a set-isomorphism from [0, 1) to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
Combining things one has that (0, 1) and S1 are set-isomorphic.
Topology is supposed to distinguish between (0, 1) and S1.
By the way, both spaces inherit a metric, (0, 1) from R, and S1
from R2.
Anything true about all metric space will therefore not beuseful when searching for a difference between the two spaces.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
Combining things one has that (0, 1) and S1 are set-isomorphic.
Topology is supposed to distinguish between (0, 1) and S1.
By the way, both spaces inherit a metric, (0, 1) from R, and S1
from R2.
Anything true about all metric space will therefore not beuseful when searching for a difference between the two spaces.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
Combining things one has that (0, 1) and S1 are set-isomorphic.
Topology is supposed to distinguish between (0, 1) and S1.
By the way, both spaces inherit a metric, (0, 1) from R, and S1
from R2.
Anything true about all metric space will therefore not beuseful when searching for a difference between the two spaces.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The point with another point.
Combining things one has that (0, 1) and S1 are set-isomorphic.
Topology is supposed to distinguish between (0, 1) and S1.
By the way, both spaces inherit a metric, (0, 1) from R, and S1
from R2.
Anything true about all metric space will therefore not beuseful when searching for a difference between the two spaces.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
Let S2 be the set of points
S2 = {(x , y , z) ∈ R3 | x2 + y2 + z2 = 1},
the so-called unit-sphere.
The straight line that emanates from (0, 0, 1) and passesthrough a point p ∈ S2 intersects the xy -coordinate plane atsome point q = (x , y , 0).
This map, stereo-graphic projection, produces aset-isomorphism from the punctured sphere to R2.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
Let S2 be the set of points
S2 = {(x , y , z) ∈ R3 | x2 + y2 + z2 = 1},
the so-called unit-sphere.
The straight line that emanates from (0, 0, 1) and passesthrough a point p ∈ S2 intersects the xy -coordinate plane atsome point q = (x , y , 0).
This map, stereo-graphic projection, produces aset-isomorphism from the punctured sphere to R2.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
Let S2 be the set of points
S2 = {(x , y , z) ∈ R3 | x2 + y2 + z2 = 1},
the so-called unit-sphere.
The straight line that emanates from (0, 0, 1) and passesthrough a point p ∈ S2 intersects the xy -coordinate plane atsome point q = (x , y , 0).
This map, stereo-graphic projection, produces aset-isomorphism from the punctured sphere to R2.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
There is another set-isomorphism from R2 to the puncturedplane.
Now send (0, 0, 1) to the point missing from R2.
This produces a set-isomorphism from S2 to R2.
S2 inherits a metric from R3.
Our intuition suggests that (0, 1) does not have a ‘hole’,whereas S1 does.
Similarly, R2 does not have a ‘hole’, whereas S2 does.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
There is another set-isomorphism from R2 to the puncturedplane.
Now send (0, 0, 1) to the point missing from R2.
This produces a set-isomorphism from S2 to R2.
S2 inherits a metric from R3.
Our intuition suggests that (0, 1) does not have a ‘hole’,whereas S1 does.
Similarly, R2 does not have a ‘hole’, whereas S2 does.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
There is another set-isomorphism from R2 to the puncturedplane.
Now send (0, 0, 1) to the point missing from R2.
This produces a set-isomorphism from S2 to R2.
S2 inherits a metric from R3.
Our intuition suggests that (0, 1) does not have a ‘hole’,whereas S1 does.
Similarly, R2 does not have a ‘hole’, whereas S2 does.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
There is another set-isomorphism from R2 to the puncturedplane.
Now send (0, 0, 1) to the point missing from R2.
This produces a set-isomorphism from S2 to R2.
S2 inherits a metric from R3.
Our intuition suggests that (0, 1) does not have a ‘hole’,whereas S1 does.
Similarly, R2 does not have a ‘hole’, whereas S2 does.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
There is another set-isomorphism from R2 to the puncturedplane.
Now send (0, 0, 1) to the point missing from R2.
This produces a set-isomorphism from S2 to R2.
S2 inherits a metric from R3.
Our intuition suggests that (0, 1) does not have a ‘hole’,whereas S1 does.
Similarly, R2 does not have a ‘hole’, whereas S2 does.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
There is another set-isomorphism from R2 to the puncturedplane.
Now send (0, 0, 1) to the point missing from R2.
This produces a set-isomorphism from S2 to R2.
S2 inherits a metric from R3.
Our intuition suggests that (0, 1) does not have a ‘hole’,whereas S1 does.
Similarly, R2 does not have a ‘hole’, whereas S2 does.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
What do we mean by a hole?
What about S1 × R, S1 × S1, S2 × R, S2 × S1, or S2 × S2?
Do they have holes, if so, of what kind?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
What do we mean by a hole?
What about S1 × R, S1 × S1, S2 × R, S2 × S1, or S2 × S2?
Do they have holes, if so, of what kind?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Adding a point to R2.
What do we mean by a hole?
What about S1 × R, S1 × S1, S2 × R, S2 × S1, or S2 × S2?
Do they have holes, if so, of what kind?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Let ε > 0 be given and consider the domain
Aε ={
(x , y) ∈ R2 | ε ≤ x2 + y2 ≤ 1}.
What is the outward flux from Aε?
It is now necessary to add the integral ∇f · N alongCε = {(x , y) ∈ R2 | x2 + y2 = ε} to the previous value.
This time one gets 1ε (x , y) · 1√
ε(−x ,−y) = − 1√
ε.
So for any ε > 0 the new integral has value −2π.
The total outward flux is therefore zero, which is in agreementwith Gauss.
The collection of points on and inside S1 differ from Aε foreach ε > 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Let ε > 0 be given and consider the domain
Aε ={
(x , y) ∈ R2 | ε ≤ x2 + y2 ≤ 1}.
What is the outward flux from Aε?
It is now necessary to add the integral ∇f · N alongCε = {(x , y) ∈ R2 | x2 + y2 = ε} to the previous value.
This time one gets 1ε (x , y) · 1√
ε(−x ,−y) = − 1√
ε.
So for any ε > 0 the new integral has value −2π.
The total outward flux is therefore zero, which is in agreementwith Gauss.
The collection of points on and inside S1 differ from Aε foreach ε > 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Let ε > 0 be given and consider the domain
Aε ={
(x , y) ∈ R2 | ε ≤ x2 + y2 ≤ 1}.
What is the outward flux from Aε?
It is now necessary to add the integral ∇f · N alongCε = {(x , y) ∈ R2 | x2 + y2 = ε} to the previous value.
This time one gets 1ε (x , y) · 1√
ε(−x ,−y) = − 1√
ε.
So for any ε > 0 the new integral has value −2π.
The total outward flux is therefore zero, which is in agreementwith Gauss.
The collection of points on and inside S1 differ from Aε foreach ε > 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Let ε > 0 be given and consider the domain
Aε ={
(x , y) ∈ R2 | ε ≤ x2 + y2 ≤ 1}.
What is the outward flux from Aε?
It is now necessary to add the integral ∇f · N alongCε = {(x , y) ∈ R2 | x2 + y2 = ε} to the previous value.
This time one gets 1ε (x , y) · 1√
ε(−x ,−y) = − 1√
ε.
So for any ε > 0 the new integral has value −2π.
The total outward flux is therefore zero, which is in agreementwith Gauss.
The collection of points on and inside S1 differ from Aε foreach ε > 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Let ε > 0 be given and consider the domain
Aε ={
(x , y) ∈ R2 | ε ≤ x2 + y2 ≤ 1}.
What is the outward flux from Aε?
It is now necessary to add the integral ∇f · N alongCε = {(x , y) ∈ R2 | x2 + y2 = ε} to the previous value.
This time one gets 1ε (x , y) · 1√
ε(−x ,−y) = − 1√
ε.
So for any ε > 0 the new integral has value −2π.
The total outward flux is therefore zero, which is in agreementwith Gauss.
The collection of points on and inside S1 differ from Aε foreach ε > 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Let ε > 0 be given and consider the domain
Aε ={
(x , y) ∈ R2 | ε ≤ x2 + y2 ≤ 1}.
What is the outward flux from Aε?
It is now necessary to add the integral ∇f · N alongCε = {(x , y) ∈ R2 | x2 + y2 = ε} to the previous value.
This time one gets 1ε (x , y) · 1√
ε(−x ,−y) = − 1√
ε.
So for any ε > 0 the new integral has value −2π.
The total outward flux is therefore zero, which is in agreementwith Gauss.
The collection of points on and inside S1 differ from Aε foreach ε > 0.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Now consider the scalar field g .
This time one gets (x , y) · 1√ε(−x ,−y) = −
√ε.
So for any ε > 0 the new integral has value −2πε.
The total outward flux is therefore 2π − 2πε, which is inagreement with Gauss since the divergence is 2 and2 · (π(1)2)− 2(π · (
√ε)2) = 2π − 2πε.
In this case the flux through Cε tends to zero as ε tends to zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Now consider the scalar field g .
This time one gets (x , y) · 1√ε(−x ,−y) = −
√ε.
So for any ε > 0 the new integral has value −2πε.
The total outward flux is therefore 2π − 2πε, which is inagreement with Gauss since the divergence is 2 and2 · (π(1)2)− 2(π · (
√ε)2) = 2π − 2πε.
In this case the flux through Cε tends to zero as ε tends to zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Now consider the scalar field g .
This time one gets (x , y) · 1√ε(−x ,−y) = −
√ε.
So for any ε > 0 the new integral has value −2πε.
The total outward flux is therefore 2π − 2πε, which is inagreement with Gauss since the divergence is 2 and2 · (π(1)2)− 2(π · (
√ε)2) = 2π − 2πε.
In this case the flux through Cε tends to zero as ε tends to zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Now consider the scalar field g .
This time one gets (x , y) · 1√ε(−x ,−y) = −
√ε.
So for any ε > 0 the new integral has value −2πε.
The total outward flux is therefore 2π − 2πε, which is inagreement with Gauss since the divergence is 2 and2 · (π(1)2)− 2(π · (
√ε)2) = 2π − 2πε.
In this case the flux through Cε tends to zero as ε tends to zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A closer look.
Now consider the scalar field g .
This time one gets (x , y) · 1√ε(−x ,−y) = −
√ε.
So for any ε > 0 the new integral has value −2πε.
The total outward flux is therefore 2π − 2πε, which is inagreement with Gauss since the divergence is 2 and2 · (π(1)2)− 2(π · (
√ε)2) = 2π − 2πε.
In this case the flux through Cε tends to zero as ε tends to zero.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open intervals.
Suppose a, b ∈ R satisfy a < b.
Write(a, b) = {x ∈ R | a < x < b} .
The context will make sure this is not confused with thenotation for a point in R2.
The set (a, b) is known as an open interval in R.
A finite intersection of open intervals is either empty or anopen interval.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open intervals.
Suppose a, b ∈ R satisfy a < b.
Write(a, b) = {x ∈ R | a < x < b} .
The context will make sure this is not confused with thenotation for a point in R2.
The set (a, b) is known as an open interval in R.
A finite intersection of open intervals is either empty or anopen interval.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open intervals.
Suppose a, b ∈ R satisfy a < b.
Write(a, b) = {x ∈ R | a < x < b} .
The context will make sure this is not confused with thenotation for a point in R2.
The set (a, b) is known as an open interval in R.
A finite intersection of open intervals is either empty or anopen interval.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open intervals.
Suppose a, b ∈ R satisfy a < b.
Write(a, b) = {x ∈ R | a < x < b} .
The context will make sure this is not confused with thenotation for a point in R2.
The set (a, b) is known as an open interval in R.
A finite intersection of open intervals is either empty or anopen interval.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open intervals.
Suppose a, b ∈ R satisfy a < b.
Write(a, b) = {x ∈ R | a < x < b} .
The context will make sure this is not confused with thenotation for a point in R2.
The set (a, b) is known as an open interval in R.
A finite intersection of open intervals is either empty or anopen interval.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Unions are not well-behaved.
The union of (0, 1) and (2, 3) is not an open interval.
The idea is to localize things.
A subset U ⊂ R is said to be open if to each x ∈ U there issome rx > 0 such that (x − rx , x + rx) ⊂ U.
It is an important technical exercise to establish that an openinterval is indeed open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Unions are not well-behaved.
The union of (0, 1) and (2, 3) is not an open interval.
The idea is to localize things.
A subset U ⊂ R is said to be open if to each x ∈ U there issome rx > 0 such that (x − rx , x + rx) ⊂ U.
It is an important technical exercise to establish that an openinterval is indeed open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Unions are not well-behaved.
The union of (0, 1) and (2, 3) is not an open interval.
The idea is to localize things.
A subset U ⊂ R is said to be open if to each x ∈ U there issome rx > 0 such that (x − rx , x + rx) ⊂ U.
It is an important technical exercise to establish that an openinterval is indeed open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Unions are not well-behaved.
The union of (0, 1) and (2, 3) is not an open interval.
The idea is to localize things.
A subset U ⊂ R is said to be open if to each x ∈ U there issome rx > 0 such that (x − rx , x + rx) ⊂ U.
It is an important technical exercise to establish that an openinterval is indeed open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Unions are not well-behaved.
The union of (0, 1) and (2, 3) is not an open interval.
The idea is to localize things.
A subset U ⊂ R is said to be open if to each x ∈ U there issome rx > 0 such that (x − rx , x + rx) ⊂ U.
It is an important technical exercise to establish that an openinterval is indeed open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Now any union of open sets in R is open.
While finite intersection of open sets in R are open,
∞⋂k=1
(−1
k,
1
k) = {0}
is not an open set.
Observe that it is important to not drop the braces.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Now any union of open sets in R is open.
While finite intersection of open sets in R are open,
∞⋂k=1
(−1
k,
1
k) = {0}
is not an open set.
Observe that it is important to not drop the braces.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in R.
Now any union of open sets in R is open.
While finite intersection of open sets in R are open,
∞⋂k=1
(−1
k,
1
k) = {0}
is not an open set.
Observe that it is important to not drop the braces.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in metric spaces.
Consider a metric space (X , d) where d is the metric.
For r > 0 let Bx(r) be given by
Bx(r) = {y ∈ X | d(y , x) < r} .
It is known as the open ball of radius r centered at x ,
A subset U ⊂ X is said to be open if to each x ∈ U there issome rx > 0 such that Bx(rx) ⊂ U.
It is again an exercise to show that an open ball indeed is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in metric spaces.
Consider a metric space (X , d) where d is the metric.
For r > 0 let Bx(r) be given by
Bx(r) = {y ∈ X | d(y , x) < r} .
It is known as the open ball of radius r centered at x ,
A subset U ⊂ X is said to be open if to each x ∈ U there issome rx > 0 such that Bx(rx) ⊂ U.
It is again an exercise to show that an open ball indeed is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in metric spaces.
Consider a metric space (X , d) where d is the metric.
For r > 0 let Bx(r) be given by
Bx(r) = {y ∈ X | d(y , x) < r} .
It is known as the open ball of radius r centered at x ,
A subset U ⊂ X is said to be open if to each x ∈ U there issome rx > 0 such that Bx(rx) ⊂ U.
It is again an exercise to show that an open ball indeed is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in metric spaces.
Consider a metric space (X , d) where d is the metric.
For r > 0 let Bx(r) be given by
Bx(r) = {y ∈ X | d(y , x) < r} .
It is known as the open ball of radius r centered at x ,
A subset U ⊂ X is said to be open if to each x ∈ U there issome rx > 0 such that Bx(rx) ⊂ U.
It is again an exercise to show that an open ball indeed is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Open sets in metric spaces.
Consider a metric space (X , d) where d is the metric.
For r > 0 let Bx(r) be given by
Bx(r) = {y ∈ X | d(y , x) < r} .
It is known as the open ball of radius r centered at x ,
A subset U ⊂ X is said to be open if to each x ∈ U there issome rx > 0 such that Bx(rx) ⊂ U.
It is again an exercise to show that an open ball indeed is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
(1) It is immediate that X and ∅ are open.
(2) Any union of open sets is open.
(3) Any finite intersection of open sets is open.
The collection of all open sets in X is known as the ‘metrictopology’ of (X , d).
There may be other collections τ of subsets in X that satisfy(1)-(3).
In this case on writes (X , τ) and τ is said to be a topology.
A simple example is given by the collection of all subsets, whichis known as the ‘discrete topology’.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
(1) It is immediate that X and ∅ are open.
(2) Any union of open sets is open.
(3) Any finite intersection of open sets is open.
The collection of all open sets in X is known as the ‘metrictopology’ of (X , d).
There may be other collections τ of subsets in X that satisfy(1)-(3).
In this case on writes (X , τ) and τ is said to be a topology.
A simple example is given by the collection of all subsets, whichis known as the ‘discrete topology’.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
(1) It is immediate that X and ∅ are open.
(2) Any union of open sets is open.
(3) Any finite intersection of open sets is open.
The collection of all open sets in X is known as the ‘metrictopology’ of (X , d).
There may be other collections τ of subsets in X that satisfy(1)-(3).
In this case on writes (X , τ) and τ is said to be a topology.
A simple example is given by the collection of all subsets, whichis known as the ‘discrete topology’.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
(1) It is immediate that X and ∅ are open.
(2) Any union of open sets is open.
(3) Any finite intersection of open sets is open.
The collection of all open sets in X is known as the ‘metrictopology’ of (X , d).
There may be other collections τ of subsets in X that satisfy(1)-(3).
In this case on writes (X , τ) and τ is said to be a topology.
A simple example is given by the collection of all subsets, whichis known as the ‘discrete topology’.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
(1) It is immediate that X and ∅ are open.
(2) Any union of open sets is open.
(3) Any finite intersection of open sets is open.
The collection of all open sets in X is known as the ‘metrictopology’ of (X , d).
There may be other collections τ of subsets in X that satisfy(1)-(3).
In this case on writes (X , τ) and τ is said to be a topology.
A simple example is given by the collection of all subsets, whichis known as the ‘discrete topology’.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
(1) It is immediate that X and ∅ are open.
(2) Any union of open sets is open.
(3) Any finite intersection of open sets is open.
The collection of all open sets in X is known as the ‘metrictopology’ of (X , d).
There may be other collections τ of subsets in X that satisfy(1)-(3).
In this case on writes (X , τ) and τ is said to be a topology.
A simple example is given by the collection of all subsets, whichis known as the ‘discrete topology’.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
(1) It is immediate that X and ∅ are open.
(2) Any union of open sets is open.
(3) Any finite intersection of open sets is open.
The collection of all open sets in X is known as the ‘metrictopology’ of (X , d).
There may be other collections τ of subsets in X that satisfy(1)-(3).
In this case on writes (X , τ) and τ is said to be a topology.
A simple example is given by the collection of all subsets, whichis known as the ‘discrete topology’.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
Another ‘extreme’ example is given by the collection {∅,X}.
This topology is known as the ‘indiscrete topology’.
An interesting question to contemplate is whether it is possibleto find a metric that produces a given topology.
In the discrete case the answer is yes.
Use
d0(x , y) =
{0 x = y ;
1 x 6= y .
In the indiscrete case the answer is no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
Another ‘extreme’ example is given by the collection {∅,X}.
This topology is known as the ‘indiscrete topology’.
An interesting question to contemplate is whether it is possibleto find a metric that produces a given topology.
In the discrete case the answer is yes.
Use
d0(x , y) =
{0 x = y ;
1 x 6= y .
In the indiscrete case the answer is no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
Another ‘extreme’ example is given by the collection {∅,X}.
This topology is known as the ‘indiscrete topology’.
An interesting question to contemplate is whether it is possibleto find a metric that produces a given topology.
In the discrete case the answer is yes.
Use
d0(x , y) =
{0 x = y ;
1 x 6= y .
In the indiscrete case the answer is no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
Another ‘extreme’ example is given by the collection {∅,X}.
This topology is known as the ‘indiscrete topology’.
An interesting question to contemplate is whether it is possibleto find a metric that produces a given topology.
In the discrete case the answer is yes.
Use
d0(x , y) =
{0 x = y ;
1 x 6= y .
In the indiscrete case the answer is no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
Another ‘extreme’ example is given by the collection {∅,X}.
This topology is known as the ‘indiscrete topology’.
An interesting question to contemplate is whether it is possibleto find a metric that produces a given topology.
In the discrete case the answer is yes.
Use
d0(x , y) =
{0 x = y ;
1 x 6= y .
In the indiscrete case the answer is no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
Another ‘extreme’ example is given by the collection {∅,X}.
This topology is known as the ‘indiscrete topology’.
An interesting question to contemplate is whether it is possibleto find a metric that produces a given topology.
In the discrete case the answer is yes.
Use
d0(x , y) =
{0 x = y ;
1 x 6= y .
In the indiscrete case the answer is no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
The discrete and indiscrete topology is available in any set,whether that set has a metric defined on it or not.
Let X = {a, b} and τ = {∅, {a},X}.
This is in fact a topology and it is known as Sierpinski’stopology.
Topologies on finite sets are known as finite topologies.
There is no metric on this X that produces Sierpinski’stopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
The discrete and indiscrete topology is available in any set,whether that set has a metric defined on it or not.
Let X = {a, b} and τ = {∅, {a},X}.
This is in fact a topology and it is known as Sierpinski’stopology.
Topologies on finite sets are known as finite topologies.
There is no metric on this X that produces Sierpinski’stopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
The discrete and indiscrete topology is available in any set,whether that set has a metric defined on it or not.
Let X = {a, b} and τ = {∅, {a},X}.
This is in fact a topology and it is known as Sierpinski’stopology.
Topologies on finite sets are known as finite topologies.
There is no metric on this X that produces Sierpinski’stopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
The discrete and indiscrete topology is available in any set,whether that set has a metric defined on it or not.
Let X = {a, b} and τ = {∅, {a},X}.
This is in fact a topology and it is known as Sierpinski’stopology.
Topologies on finite sets are known as finite topologies.
There is no metric on this X that produces Sierpinski’stopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Topology.
The discrete and indiscrete topology is available in any set,whether that set has a metric defined on it or not.
Let X = {a, b} and τ = {∅, {a},X}.
This is in fact a topology and it is known as Sierpinski’stopology.
Topologies on finite sets are known as finite topologies.
There is no metric on this X that produces Sierpinski’stopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially the same topologies.
Now let τ0 = {∅, {b},X}.
This is also a topology, and it is ‘essentially’ the same as τ .
To make this precise consider the map given by φ(a) = b andφ(b) = a.
This map is a set-isomorphism.
But more is true, it takes open sets to open sets and theinverse does too.
A map with these properties is said to be a homeomorphismfrom (X , τ) to (X , τ0).
The two spaces involved are said to be homeomorphic.
Much of topology concerns settling whether two given spacesare homeomorphic or not.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Homeomorphisms.
It is in general not trivial to produce a set-isomorphism so itshould come as no surprise that it is even more difficult toproduce a homeomorphism.
To rule out the existence of a homeomorphism when the spacesare not homeomorphic also seems like a daunting task.
An idea that will be helpful is to search for properties thatmust be shared by homeomorphic spaces.
A property with this feature is known as a topological property.
An indication of how this may work remove any point from theinterval (0, 1) and observe how this produces two intervals.
Now remove a point from S1, and observe that this will neverproduce two pieces.
This suggests that (0, 1) and S1 are not homeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Homeomorphisms.
It is in general not trivial to produce a set-isomorphism so itshould come as no surprise that it is even more difficult toproduce a homeomorphism.
To rule out the existence of a homeomorphism when the spacesare not homeomorphic also seems like a daunting task.
An idea that will be helpful is to search for properties thatmust be shared by homeomorphic spaces.
A property with this feature is known as a topological property.
An indication of how this may work remove any point from theinterval (0, 1) and observe how this produces two intervals.
Now remove a point from S1, and observe that this will neverproduce two pieces.
This suggests that (0, 1) and S1 are not homeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Homeomorphisms.
It is in general not trivial to produce a set-isomorphism so itshould come as no surprise that it is even more difficult toproduce a homeomorphism.
To rule out the existence of a homeomorphism when the spacesare not homeomorphic also seems like a daunting task.
An idea that will be helpful is to search for properties thatmust be shared by homeomorphic spaces.
A property with this feature is known as a topological property.
An indication of how this may work remove any point from theinterval (0, 1) and observe how this produces two intervals.
Now remove a point from S1, and observe that this will neverproduce two pieces.
This suggests that (0, 1) and S1 are not homeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Homeomorphisms.
It is in general not trivial to produce a set-isomorphism so itshould come as no surprise that it is even more difficult toproduce a homeomorphism.
To rule out the existence of a homeomorphism when the spacesare not homeomorphic also seems like a daunting task.
An idea that will be helpful is to search for properties thatmust be shared by homeomorphic spaces.
A property with this feature is known as a topological property.
An indication of how this may work remove any point from theinterval (0, 1) and observe how this produces two intervals.
Now remove a point from S1, and observe that this will neverproduce two pieces.
This suggests that (0, 1) and S1 are not homeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Homeomorphisms.
It is in general not trivial to produce a set-isomorphism so itshould come as no surprise that it is even more difficult toproduce a homeomorphism.
To rule out the existence of a homeomorphism when the spacesare not homeomorphic also seems like a daunting task.
An idea that will be helpful is to search for properties thatmust be shared by homeomorphic spaces.
A property with this feature is known as a topological property.
An indication of how this may work remove any point from theinterval (0, 1) and observe how this produces two intervals.
Now remove a point from S1, and observe that this will neverproduce two pieces.
This suggests that (0, 1) and S1 are not homeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Homeomorphisms.
It is in general not trivial to produce a set-isomorphism so itshould come as no surprise that it is even more difficult toproduce a homeomorphism.
To rule out the existence of a homeomorphism when the spacesare not homeomorphic also seems like a daunting task.
An idea that will be helpful is to search for properties thatmust be shared by homeomorphic spaces.
A property with this feature is known as a topological property.
An indication of how this may work remove any point from theinterval (0, 1) and observe how this produces two intervals.
Now remove a point from S1, and observe that this will neverproduce two pieces.
This suggests that (0, 1) and S1 are not homeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Homeomorphisms.
It is in general not trivial to produce a set-isomorphism so itshould come as no surprise that it is even more difficult toproduce a homeomorphism.
To rule out the existence of a homeomorphism when the spacesare not homeomorphic also seems like a daunting task.
An idea that will be helpful is to search for properties thatmust be shared by homeomorphic spaces.
A property with this feature is known as a topological property.
An indication of how this may work remove any point from theinterval (0, 1) and observe how this produces two intervals.
Now remove a point from S1, and observe that this will neverproduce two pieces.
This suggests that (0, 1) and S1 are not homeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
A topology specifies which subsets are considered open inaddition to the empty set and the space itself.
The complement of an open set is known as a closed set.
Both the empty set and the space itself are always closed.
This also illustrates that the concepts of open and closed arenot complementary.
The presence of some subset other than the empty set and thewhole space that is both open and closed is a topologicalproperty.
The absence of such a set in any indiscrete topology and thepresence of many such sets in any discrete topology confirmsthe suspicion that no indiscrete and discrete topologies arehomeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
A topology specifies which subsets are considered open inaddition to the empty set and the space itself.
The complement of an open set is known as a closed set.
Both the empty set and the space itself are always closed.
This also illustrates that the concepts of open and closed arenot complementary.
The presence of some subset other than the empty set and thewhole space that is both open and closed is a topologicalproperty.
The absence of such a set in any indiscrete topology and thepresence of many such sets in any discrete topology confirmsthe suspicion that no indiscrete and discrete topologies arehomeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
A topology specifies which subsets are considered open inaddition to the empty set and the space itself.
The complement of an open set is known as a closed set.
Both the empty set and the space itself are always closed.
This also illustrates that the concepts of open and closed arenot complementary.
The presence of some subset other than the empty set and thewhole space that is both open and closed is a topologicalproperty.
The absence of such a set in any indiscrete topology and thepresence of many such sets in any discrete topology confirmsthe suspicion that no indiscrete and discrete topologies arehomeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
A topology specifies which subsets are considered open inaddition to the empty set and the space itself.
The complement of an open set is known as a closed set.
Both the empty set and the space itself are always closed.
This also illustrates that the concepts of open and closed arenot complementary.
The presence of some subset other than the empty set and thewhole space that is both open and closed is a topologicalproperty.
The absence of such a set in any indiscrete topology and thepresence of many such sets in any discrete topology confirmsthe suspicion that no indiscrete and discrete topologies arehomeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
A topology specifies which subsets are considered open inaddition to the empty set and the space itself.
The complement of an open set is known as a closed set.
Both the empty set and the space itself are always closed.
This also illustrates that the concepts of open and closed arenot complementary.
The presence of some subset other than the empty set and thewhole space that is both open and closed is a topologicalproperty.
The absence of such a set in any indiscrete topology and thepresence of many such sets in any discrete topology confirmsthe suspicion that no indiscrete and discrete topologies arehomeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
A topology specifies which subsets are considered open inaddition to the empty set and the space itself.
The complement of an open set is known as a closed set.
Both the empty set and the space itself are always closed.
This also illustrates that the concepts of open and closed arenot complementary.
The presence of some subset other than the empty set and thewhole space that is both open and closed is a topologicalproperty.
The absence of such a set in any indiscrete topology and thepresence of many such sets in any discrete topology confirmsthe suspicion that no indiscrete and discrete topologies arehomeomorphic.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
Using basic set operations one concludes that any intersectionof closed sets must be closed.
Any finite union of closed sets must also be closed.
The union∞⋃k=2
[0, 1− 1
k] = [0, 1)
shows that arbitrary unions of closed sets need not be closed.
Subsets with a single point are known as singletons.
The indiscrete and Sierpinski’s topologies show that singletonsneed not be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
Using basic set operations one concludes that any intersectionof closed sets must be closed.
Any finite union of closed sets must also be closed.
The union∞⋃k=2
[0, 1− 1
k] = [0, 1)
shows that arbitrary unions of closed sets need not be closed.
Subsets with a single point are known as singletons.
The indiscrete and Sierpinski’s topologies show that singletonsneed not be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
Using basic set operations one concludes that any intersectionof closed sets must be closed.
Any finite union of closed sets must also be closed.
The union∞⋃k=2
[0, 1− 1
k] = [0, 1)
shows that arbitrary unions of closed sets need not be closed.
Subsets with a single point are known as singletons.
The indiscrete and Sierpinski’s topologies show that singletonsneed not be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
Using basic set operations one concludes that any intersectionof closed sets must be closed.
Any finite union of closed sets must also be closed.
The union∞⋃k=2
[0, 1− 1
k] = [0, 1)
shows that arbitrary unions of closed sets need not be closed.
Subsets with a single point are known as singletons.
The indiscrete and Sierpinski’s topologies show that singletonsneed not be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closed sets.
Using basic set operations one concludes that any intersectionof closed sets must be closed.
Any finite union of closed sets must also be closed.
The union∞⋃k=2
[0, 1− 1
k] = [0, 1)
shows that arbitrary unions of closed sets need not be closed.
Subsets with a single point are known as singletons.
The indiscrete and Sierpinski’s topologies show that singletonsneed not be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
For the discussion to be meaningful it will be assumed that thespace has at least two points.
To get started let X = {a, b}.
There is a total of 22 = 4 subsets available.
To have a topology, ∅ and X must be included.
Next one must decide whether to include any of the twosingletons.
There are 2 · 2 = 4 possible decisions.
It turns out the all the 4 possibilities produce a topology.
In summary one has the indiscrete, two Sierpinski topologies,and the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Since the two Sierpinski topologies are homeomorphic, theyshould only be counted as one.
In the collection of all topologies available in a space define twotopologies to be equivalent whenever they are homeomorphic.
This produces an equivalence relation and the count that reallyis of interest is the number of equivalence classes.
In the present case this count is 3.
A little bit of work remains to establish that the indiscretetopology is not homeomorphic to Sierpinski’s topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Since the two Sierpinski topologies are homeomorphic, theyshould only be counted as one.
In the collection of all topologies available in a space define twotopologies to be equivalent whenever they are homeomorphic.
This produces an equivalence relation and the count that reallyis of interest is the number of equivalence classes.
In the present case this count is 3.
A little bit of work remains to establish that the indiscretetopology is not homeomorphic to Sierpinski’s topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Since the two Sierpinski topologies are homeomorphic, theyshould only be counted as one.
In the collection of all topologies available in a space define twotopologies to be equivalent whenever they are homeomorphic.
This produces an equivalence relation and the count that reallyis of interest is the number of equivalence classes.
In the present case this count is 3.
A little bit of work remains to establish that the indiscretetopology is not homeomorphic to Sierpinski’s topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Since the two Sierpinski topologies are homeomorphic, theyshould only be counted as one.
In the collection of all topologies available in a space define twotopologies to be equivalent whenever they are homeomorphic.
This produces an equivalence relation and the count that reallyis of interest is the number of equivalence classes.
In the present case this count is 3.
A little bit of work remains to establish that the indiscretetopology is not homeomorphic to Sierpinski’s topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Since the two Sierpinski topologies are homeomorphic, theyshould only be counted as one.
In the collection of all topologies available in a space define twotopologies to be equivalent whenever they are homeomorphic.
This produces an equivalence relation and the count that reallyis of interest is the number of equivalence classes.
In the present case this count is 3.
A little bit of work remains to establish that the indiscretetopology is not homeomorphic to Sierpinski’s topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Now consider X = {a, b, c}.
There is a total of 23 = 8 subsets available.
This means that this time there are 6 choices with 26 = 64possible decisions.
Ignore the possibilities of choosing none or all to reduce it to62.
It turns out that out of the remaining 62 there are 27 that arein fact topologies,
A deeper analysis reveals that in the end there are 9 essentiallydifferent topologies on a 3-point space.
Other than the indiscrete and the discrete they are representedby
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Now consider X = {a, b, c}.
There is a total of 23 = 8 subsets available.
This means that this time there are 6 choices with 26 = 64possible decisions.
Ignore the possibilities of choosing none or all to reduce it to62.
It turns out that out of the remaining 62 there are 27 that arein fact topologies,
A deeper analysis reveals that in the end there are 9 essentiallydifferent topologies on a 3-point space.
Other than the indiscrete and the discrete they are representedby
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Now consider X = {a, b, c}.
There is a total of 23 = 8 subsets available.
This means that this time there are 6 choices with 26 = 64possible decisions.
Ignore the possibilities of choosing none or all to reduce it to62.
It turns out that out of the remaining 62 there are 27 that arein fact topologies,
A deeper analysis reveals that in the end there are 9 essentiallydifferent topologies on a 3-point space.
Other than the indiscrete and the discrete they are representedby
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Now consider X = {a, b, c}.
There is a total of 23 = 8 subsets available.
This means that this time there are 6 choices with 26 = 64possible decisions.
Ignore the possibilities of choosing none or all to reduce it to62.
It turns out that out of the remaining 62 there are 27 that arein fact topologies,
A deeper analysis reveals that in the end there are 9 essentiallydifferent topologies on a 3-point space.
Other than the indiscrete and the discrete they are representedby
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Now consider X = {a, b, c}.
There is a total of 23 = 8 subsets available.
This means that this time there are 6 choices with 26 = 64possible decisions.
Ignore the possibilities of choosing none or all to reduce it to62.
It turns out that out of the remaining 62 there are 27 that arein fact topologies,
A deeper analysis reveals that in the end there are 9 essentiallydifferent topologies on a 3-point space.
Other than the indiscrete and the discrete they are representedby
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Now consider X = {a, b, c}.
There is a total of 23 = 8 subsets available.
This means that this time there are 6 choices with 26 = 64possible decisions.
Ignore the possibilities of choosing none or all to reduce it to62.
It turns out that out of the remaining 62 there are 27 that arein fact topologies,
A deeper analysis reveals that in the end there are 9 essentiallydifferent topologies on a 3-point space.
Other than the indiscrete and the discrete they are representedby
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Counting finite topologies.
Now consider X = {a, b, c}.
There is a total of 23 = 8 subsets available.
This means that this time there are 6 choices with 26 = 64possible decisions.
Ignore the possibilities of choosing none or all to reduce it to62.
It turns out that out of the remaining 62 there are 27 that arein fact topologies,
A deeper analysis reveals that in the end there are 9 essentiallydifferent topologies on a 3-point space.
Other than the indiscrete and the discrete they are representedby
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Essentially different 3-point topologies.
a
ab
aab
abc
a bab
aab
ac
a bab
ac
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
What happens when the number of pointsincreases?
The general sequence Nk , where k is the number of points,may be written as N2 = 3, N3 = 9, N4 = 33, N5 = 139,N6 = 718, N7 = 4535, N8 = 35979, N9 = 363083,N10 = 4717687, N11 = 79501654, N12 = 1744252509,N13 = 49872339897, N14 = 1856792610995,N15 = 89847422244493, and N16 = 5637294117525695.
The values of Nk when k ≥ 17 are not known and represent avery stiff computational challenge.
The web-based ‘On-line encyclopedia of integer sequences’tracks these kind of ‘world-records’.
Apparently there is a vast number of topologies to ponder.
Almost all topologies that appear in practice are related tosome other mathematical structure.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
What happens when the number of pointsincreases?
The general sequence Nk , where k is the number of points,may be written as N2 = 3, N3 = 9, N4 = 33, N5 = 139,N6 = 718, N7 = 4535, N8 = 35979, N9 = 363083,N10 = 4717687, N11 = 79501654, N12 = 1744252509,N13 = 49872339897, N14 = 1856792610995,N15 = 89847422244493, and N16 = 5637294117525695.
The values of Nk when k ≥ 17 are not known and represent avery stiff computational challenge.
The web-based ‘On-line encyclopedia of integer sequences’tracks these kind of ‘world-records’.
Apparently there is a vast number of topologies to ponder.
Almost all topologies that appear in practice are related tosome other mathematical structure.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
What happens when the number of pointsincreases?
The general sequence Nk , where k is the number of points,may be written as N2 = 3, N3 = 9, N4 = 33, N5 = 139,N6 = 718, N7 = 4535, N8 = 35979, N9 = 363083,N10 = 4717687, N11 = 79501654, N12 = 1744252509,N13 = 49872339897, N14 = 1856792610995,N15 = 89847422244493, and N16 = 5637294117525695.
The values of Nk when k ≥ 17 are not known and represent avery stiff computational challenge.
The web-based ‘On-line encyclopedia of integer sequences’tracks these kind of ‘world-records’.
Apparently there is a vast number of topologies to ponder.
Almost all topologies that appear in practice are related tosome other mathematical structure.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
What happens when the number of pointsincreases?
The general sequence Nk , where k is the number of points,may be written as N2 = 3, N3 = 9, N4 = 33, N5 = 139,N6 = 718, N7 = 4535, N8 = 35979, N9 = 363083,N10 = 4717687, N11 = 79501654, N12 = 1744252509,N13 = 49872339897, N14 = 1856792610995,N15 = 89847422244493, and N16 = 5637294117525695.
The values of Nk when k ≥ 17 are not known and represent avery stiff computational challenge.
The web-based ‘On-line encyclopedia of integer sequences’tracks these kind of ‘world-records’.
Apparently there is a vast number of topologies to ponder.
Almost all topologies that appear in practice are related tosome other mathematical structure.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
What happens when the number of pointsincreases?
The general sequence Nk , where k is the number of points,may be written as N2 = 3, N3 = 9, N4 = 33, N5 = 139,N6 = 718, N7 = 4535, N8 = 35979, N9 = 363083,N10 = 4717687, N11 = 79501654, N12 = 1744252509,N13 = 49872339897, N14 = 1856792610995,N15 = 89847422244493, and N16 = 5637294117525695.
The values of Nk when k ≥ 17 are not known and represent avery stiff computational challenge.
The web-based ‘On-line encyclopedia of integer sequences’tracks these kind of ‘world-records’.
Apparently there is a vast number of topologies to ponder.
Almost all topologies that appear in practice are related tosome other mathematical structure.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
If a point x belongs to an open set U, then U is said to be aneighborhood of x , or more precisely, an open neighborhood ofx .
If x ∈ N, then N is a neighborhood of x if there is an open setU ⊂ N such that x ∈ U.
A pair of points x , y ∈ X are said to be topologicallydistinguishable if there is an open set that contains one of thepoints but not the other.
There is no pair of topologically distinguishable points in theindiscrete topology.
Sierpinski’s topology has a pair of topologically distinguishablepoints so it is therefore not homeomorphic to the indiscretetopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
If a point x belongs to an open set U, then U is said to be aneighborhood of x , or more precisely, an open neighborhood ofx .
If x ∈ N, then N is a neighborhood of x if there is an open setU ⊂ N such that x ∈ U.
A pair of points x , y ∈ X are said to be topologicallydistinguishable if there is an open set that contains one of thepoints but not the other.
There is no pair of topologically distinguishable points in theindiscrete topology.
Sierpinski’s topology has a pair of topologically distinguishablepoints so it is therefore not homeomorphic to the indiscretetopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
If a point x belongs to an open set U, then U is said to be aneighborhood of x , or more precisely, an open neighborhood ofx .
If x ∈ N, then N is a neighborhood of x if there is an open setU ⊂ N such that x ∈ U.
A pair of points x , y ∈ X are said to be topologicallydistinguishable if there is an open set that contains one of thepoints but not the other.
There is no pair of topologically distinguishable points in theindiscrete topology.
Sierpinski’s topology has a pair of topologically distinguishablepoints so it is therefore not homeomorphic to the indiscretetopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
If a point x belongs to an open set U, then U is said to be aneighborhood of x , or more precisely, an open neighborhood ofx .
If x ∈ N, then N is a neighborhood of x if there is an open setU ⊂ N such that x ∈ U.
A pair of points x , y ∈ X are said to be topologicallydistinguishable if there is an open set that contains one of thepoints but not the other.
There is no pair of topologically distinguishable points in theindiscrete topology.
Sierpinski’s topology has a pair of topologically distinguishablepoints so it is therefore not homeomorphic to the indiscretetopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
If a point x belongs to an open set U, then U is said to be aneighborhood of x , or more precisely, an open neighborhood ofx .
If x ∈ N, then N is a neighborhood of x if there is an open setU ⊂ N such that x ∈ U.
A pair of points x , y ∈ X are said to be topologicallydistinguishable if there is an open set that contains one of thepoints but not the other.
There is no pair of topologically distinguishable points in theindiscrete topology.
Sierpinski’s topology has a pair of topologically distinguishablepoints so it is therefore not homeomorphic to the indiscretetopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
A pair of points x , y ∈ X are said to be separated if each iscontained in an open set that does not contain the other point.
If any pair of distinct points in a space are topologicallydistinguishable, then the space is said to be Kolmogorov, or T0.
If any pair of distinct points in a space are separated, then thespace is said to be Frechet, or T1.
Unfortunately, this name is also used in functional analysis fortotally unrelated spaces.
Sierpinski’s topology is T0 but not T1.
A space (X , τ) is Hausdorff, or T2, if any distinct pair of pointsx , y ∈ X has a pair of open sets Ux ,Vy ∈ τ with Ux ∩ Vy = ∅.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
A pair of points x , y ∈ X are said to be separated if each iscontained in an open set that does not contain the other point.
If any pair of distinct points in a space are topologicallydistinguishable, then the space is said to be Kolmogorov, or T0.
If any pair of distinct points in a space are separated, then thespace is said to be Frechet, or T1.
Unfortunately, this name is also used in functional analysis fortotally unrelated spaces.
Sierpinski’s topology is T0 but not T1.
A space (X , τ) is Hausdorff, or T2, if any distinct pair of pointsx , y ∈ X has a pair of open sets Ux ,Vy ∈ τ with Ux ∩ Vy = ∅.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
A pair of points x , y ∈ X are said to be separated if each iscontained in an open set that does not contain the other point.
If any pair of distinct points in a space are topologicallydistinguishable, then the space is said to be Kolmogorov, or T0.
If any pair of distinct points in a space are separated, then thespace is said to be Frechet, or T1.
Unfortunately, this name is also used in functional analysis fortotally unrelated spaces.
Sierpinski’s topology is T0 but not T1.
A space (X , τ) is Hausdorff, or T2, if any distinct pair of pointsx , y ∈ X has a pair of open sets Ux ,Vy ∈ τ with Ux ∩ Vy = ∅.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
A pair of points x , y ∈ X are said to be separated if each iscontained in an open set that does not contain the other point.
If any pair of distinct points in a space are topologicallydistinguishable, then the space is said to be Kolmogorov, or T0.
If any pair of distinct points in a space are separated, then thespace is said to be Frechet, or T1.
Unfortunately, this name is also used in functional analysis fortotally unrelated spaces.
Sierpinski’s topology is T0 but not T1.
A space (X , τ) is Hausdorff, or T2, if any distinct pair of pointsx , y ∈ X has a pair of open sets Ux ,Vy ∈ τ with Ux ∩ Vy = ∅.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
A pair of points x , y ∈ X are said to be separated if each iscontained in an open set that does not contain the other point.
If any pair of distinct points in a space are topologicallydistinguishable, then the space is said to be Kolmogorov, or T0.
If any pair of distinct points in a space are separated, then thespace is said to be Frechet, or T1.
Unfortunately, this name is also used in functional analysis fortotally unrelated spaces.
Sierpinski’s topology is T0 but not T1.
A space (X , τ) is Hausdorff, or T2, if any distinct pair of pointsx , y ∈ X has a pair of open sets Ux ,Vy ∈ τ with Ux ∩ Vy = ∅.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
A pair of points x , y ∈ X are said to be separated if each iscontained in an open set that does not contain the other point.
If any pair of distinct points in a space are topologicallydistinguishable, then the space is said to be Kolmogorov, or T0.
If any pair of distinct points in a space are separated, then thespace is said to be Frechet, or T1.
Unfortunately, this name is also used in functional analysis fortotally unrelated spaces.
Sierpinski’s topology is T0 but not T1.
A space (X , τ) is Hausdorff, or T2, if any distinct pair of pointsx , y ∈ X has a pair of open sets Ux ,Vy ∈ τ with Ux ∩ Vy = ∅.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
Given a point x in a T1 space, collect all the open set Uy for allother points y with y ∈ Uy and x /∈ Uy .
The union of all the Uy is open and its complement is {x}.
It follows that all singletons are closed in a T1 space.
In a metric space
Bx(1
2d(x , y)) ∩ By (
1
2d(x , y)) = ∅.
It follows that metric spaces are Hausdorff.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
Given a point x in a T1 space, collect all the open set Uy for allother points y with y ∈ Uy and x /∈ Uy .
The union of all the Uy is open and its complement is {x}.
It follows that all singletons are closed in a T1 space.
In a metric space
Bx(1
2d(x , y)) ∩ By (
1
2d(x , y)) = ∅.
It follows that metric spaces are Hausdorff.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
Given a point x in a T1 space, collect all the open set Uy for allother points y with y ∈ Uy and x /∈ Uy .
The union of all the Uy is open and its complement is {x}.
It follows that all singletons are closed in a T1 space.
In a metric space
Bx(1
2d(x , y)) ∩ By (
1
2d(x , y)) = ∅.
It follows that metric spaces are Hausdorff.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
Given a point x in a T1 space, collect all the open set Uy for allother points y with y ∈ Uy and x /∈ Uy .
The union of all the Uy is open and its complement is {x}.
It follows that all singletons are closed in a T1 space.
In a metric space
Bx(1
2d(x , y)) ∩ By (
1
2d(x , y)) = ∅.
It follows that metric spaces are Hausdorff.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
Given a point x in a T1 space, collect all the open set Uy for allother points y with y ∈ Uy and x /∈ Uy .
The union of all the Uy is open and its complement is {x}.
It follows that all singletons are closed in a T1 space.
In a metric space
Bx(1
2d(x , y)) ∩ By (
1
2d(x , y)) = ∅.
It follows that metric spaces are Hausdorff.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
After some thought it is possible to conclude that if allsingletons are closed in a finite space, then the space must beT2.
Moreover, the only finite topology that is T1 is the discrete.
It follows that there is a vast collection of finite topologies thatare not metrizable.
The increasing index in the T -notation indicates ever improvingabilities to separate things.
One idea that is pursued is to replace individual points byclosed sets.
This introduces a degree of uncertainty since singletons neednot be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
After some thought it is possible to conclude that if allsingletons are closed in a finite space, then the space must beT2.
Moreover, the only finite topology that is T1 is the discrete.
It follows that there is a vast collection of finite topologies thatare not metrizable.
The increasing index in the T -notation indicates ever improvingabilities to separate things.
One idea that is pursued is to replace individual points byclosed sets.
This introduces a degree of uncertainty since singletons neednot be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
After some thought it is possible to conclude that if allsingletons are closed in a finite space, then the space must beT2.
Moreover, the only finite topology that is T1 is the discrete.
It follows that there is a vast collection of finite topologies thatare not metrizable.
The increasing index in the T -notation indicates ever improvingabilities to separate things.
One idea that is pursued is to replace individual points byclosed sets.
This introduces a degree of uncertainty since singletons neednot be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
After some thought it is possible to conclude that if allsingletons are closed in a finite space, then the space must beT2.
Moreover, the only finite topology that is T1 is the discrete.
It follows that there is a vast collection of finite topologies thatare not metrizable.
The increasing index in the T -notation indicates ever improvingabilities to separate things.
One idea that is pursued is to replace individual points byclosed sets.
This introduces a degree of uncertainty since singletons neednot be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
After some thought it is possible to conclude that if allsingletons are closed in a finite space, then the space must beT2.
Moreover, the only finite topology that is T1 is the discrete.
It follows that there is a vast collection of finite topologies thatare not metrizable.
The increasing index in the T -notation indicates ever improvingabilities to separate things.
One idea that is pursued is to replace individual points byclosed sets.
This introduces a degree of uncertainty since singletons neednot be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Separation.
After some thought it is possible to conclude that if allsingletons are closed in a finite space, then the space must beT2.
Moreover, the only finite topology that is T1 is the discrete.
It follows that there is a vast collection of finite topologies thatare not metrizable.
The increasing index in the T -notation indicates ever improvingabilities to separate things.
One idea that is pursued is to replace individual points byclosed sets.
This introduces a degree of uncertainty since singletons neednot be closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Recall that in each topological space the empty set is bothopen and closed and so is the space itself.
There may other subsets in a topological space that are bothopen and closed.
The discrete topology on a finite set has each subset both openand closed.
A space that, like the indiscrete topology, has no subset otherthan the empty set and the whole space that is both open andclosed is said to be connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Recall that in each topological space the empty set is bothopen and closed and so is the space itself.
There may other subsets in a topological space that are bothopen and closed.
The discrete topology on a finite set has each subset both openand closed.
A space that, like the indiscrete topology, has no subset otherthan the empty set and the whole space that is both open andclosed is said to be connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Recall that in each topological space the empty set is bothopen and closed and so is the space itself.
There may other subsets in a topological space that are bothopen and closed.
The discrete topology on a finite set has each subset both openand closed.
A space that, like the indiscrete topology, has no subset otherthan the empty set and the whole space that is both open andclosed is said to be connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Recall that in each topological space the empty set is bothopen and closed and so is the space itself.
There may other subsets in a topological space that are bothopen and closed.
The discrete topology on a finite set has each subset both openand closed.
A space that, like the indiscrete topology, has no subset otherthan the empty set and the whole space that is both open andclosed is said to be connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Suppose U ⊂ R with both U and Uc open.
Assume U 6= ∅ and U 6= R.
There is some a ∈ R such that a /∈ U.
Consider the sets A = {x ∈ R | x ∈ U and x < a} andB = {x ∈ R | x ∈ U and x > a}.
Both of these sets cannot be empty since this implies U = {a}.
Assume A 6= ∅.
Since A is bounded above by a, there is a least upper bound α.
If α ∈ Uc , then (α− ε, α + ε) ⊂ Uc for some ε > 0 becauseUc is open.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
But this contradicts the least upper bound property of α.
If α ∈ U, then (α− ε, α + ε) ⊂ U for some ε > 0 because U isopen.
But then α is not an upper bound so again a contradiction.
Assume instead that B 6= ∅.
B is bounded below by a.
Let β be the greatest lower bound of B.
The previous issues are still present whether β ∈ U or β ∈ Uc .
It follows that no such U exists and hence R is connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Variations of this reasoning prove that rays and intervals,whether open, closed, or mixed, are in fact connected.
A connected set in R must in fact be of this kind.
The concept of a space being connected does not coexist wellwith the set operations.
Unions, intersections and complements will not in generalcontinue to be connected when the spaces involved areconnected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Variations of this reasoning prove that rays and intervals,whether open, closed, or mixed, are in fact connected.
A connected set in R must in fact be of this kind.
The concept of a space being connected does not coexist wellwith the set operations.
Unions, intersections and complements will not in generalcontinue to be connected when the spaces involved areconnected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Variations of this reasoning prove that rays and intervals,whether open, closed, or mixed, are in fact connected.
A connected set in R must in fact be of this kind.
The concept of a space being connected does not coexist wellwith the set operations.
Unions, intersections and complements will not in generalcontinue to be connected when the spaces involved areconnected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected spaces.
Variations of this reasoning prove that rays and intervals,whether open, closed, or mixed, are in fact connected.
A connected set in R must in fact be of this kind.
The concept of a space being connected does not coexist wellwith the set operations.
Unions, intersections and complements will not in generalcontinue to be connected when the spaces involved areconnected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures.
Every subset of a topological space is contained in a closed set,namely the space itself.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | Bc ∈ τ, A ⊂ B} .
The intersection of this family’s members is a closed set thatcontains A and is denoted by A .
One may also write A− for the closure.
Observe that A−− = A−, and if C ⊂ X is a closed set, thenC− = C .
The set ∂A = A− ∩ Ac− is a closed set known as thetopological boundary of A.
If A ⊂ X satisfies A− = X , then A is said to be dense in X .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures.
Every subset of a topological space is contained in a closed set,namely the space itself.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | Bc ∈ τ, A ⊂ B} .
The intersection of this family’s members is a closed set thatcontains A and is denoted by A .
One may also write A− for the closure.
Observe that A−− = A−, and if C ⊂ X is a closed set, thenC− = C .
The set ∂A = A− ∩ Ac− is a closed set known as thetopological boundary of A.
If A ⊂ X satisfies A− = X , then A is said to be dense in X .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures.
Every subset of a topological space is contained in a closed set,namely the space itself.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | Bc ∈ τ, A ⊂ B} .
The intersection of this family’s members is a closed set thatcontains A and is denoted by A .
One may also write A− for the closure.
Observe that A−− = A−, and if C ⊂ X is a closed set, thenC− = C .
The set ∂A = A− ∩ Ac− is a closed set known as thetopological boundary of A.
If A ⊂ X satisfies A− = X , then A is said to be dense in X .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures.
Every subset of a topological space is contained in a closed set,namely the space itself.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | Bc ∈ τ, A ⊂ B} .
The intersection of this family’s members is a closed set thatcontains A and is denoted by A .
One may also write A− for the closure.
Observe that A−− = A−, and if C ⊂ X is a closed set, thenC− = C .
The set ∂A = A− ∩ Ac− is a closed set known as thetopological boundary of A.
If A ⊂ X satisfies A− = X , then A is said to be dense in X .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures.
Every subset of a topological space is contained in a closed set,namely the space itself.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | Bc ∈ τ, A ⊂ B} .
The intersection of this family’s members is a closed set thatcontains A and is denoted by A .
One may also write A− for the closure.
Observe that A−− = A−, and if C ⊂ X is a closed set, thenC− = C .
The set ∂A = A− ∩ Ac− is a closed set known as thetopological boundary of A.
If A ⊂ X satisfies A− = X , then A is said to be dense in X .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures.
Every subset of a topological space is contained in a closed set,namely the space itself.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | Bc ∈ τ, A ⊂ B} .
The intersection of this family’s members is a closed set thatcontains A and is denoted by A .
One may also write A− for the closure.
Observe that A−− = A−, and if C ⊂ X is a closed set, thenC− = C .
The set ∂A = A− ∩ Ac− is a closed set known as thetopological boundary of A.
If A ⊂ X satisfies A− = X , then A is said to be dense in X .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures.
Every subset of a topological space is contained in a closed set,namely the space itself.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | Bc ∈ τ, A ⊂ B} .
The intersection of this family’s members is a closed set thatcontains A and is denoted by A .
One may also write A− for the closure.
Observe that A−− = A−, and if C ⊂ X is a closed set, thenC− = C .
The set ∂A = A− ∩ Ac− is a closed set known as thetopological boundary of A.
If A ⊂ X satisfies A− = X , then A is said to be dense in X .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Interiors.
Every subset of a topological space contains an open set,namely the empty set.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | B ∈ τ, B ⊂ A} .
The union of this family’s members is an open set contained inA and is denoted by A◦.
Observe that A◦◦ = A◦, and if C ∈ τ , then C ◦ = C .
Given any A ⊂ (X , τ) the space X partitions as
X = A◦ ∪ ∂A ∪ Ac◦.
Note that Q◦ = ∅, Q− = R, and ∂Q = R.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Interiors.
Every subset of a topological space contains an open set,namely the empty set.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | B ∈ τ, B ⊂ A} .
The union of this family’s members is an open set contained inA and is denoted by A◦.
Observe that A◦◦ = A◦, and if C ∈ τ , then C ◦ = C .
Given any A ⊂ (X , τ) the space X partitions as
X = A◦ ∪ ∂A ∪ Ac◦.
Note that Q◦ = ∅, Q− = R, and ∂Q = R.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Interiors.
Every subset of a topological space contains an open set,namely the empty set.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | B ∈ τ, B ⊂ A} .
The union of this family’s members is an open set contained inA and is denoted by A◦.
Observe that A◦◦ = A◦, and if C ∈ τ , then C ◦ = C .
Given any A ⊂ (X , τ) the space X partitions as
X = A◦ ∪ ∂A ∪ Ac◦.
Note that Q◦ = ∅, Q− = R, and ∂Q = R.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Interiors.
Every subset of a topological space contains an open set,namely the empty set.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | B ∈ τ, B ⊂ A} .
The union of this family’s members is an open set contained inA and is denoted by A◦.
Observe that A◦◦ = A◦, and if C ∈ τ , then C ◦ = C .
Given any A ⊂ (X , τ) the space X partitions as
X = A◦ ∪ ∂A ∪ Ac◦.
Note that Q◦ = ∅, Q− = R, and ∂Q = R.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Interiors.
Every subset of a topological space contains an open set,namely the empty set.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | B ∈ τ, B ⊂ A} .
The union of this family’s members is an open set contained inA and is denoted by A◦.
Observe that A◦◦ = A◦, and if C ∈ τ , then C ◦ = C .
Given any A ⊂ (X , τ) the space X partitions as
X = A◦ ∪ ∂A ∪ Ac◦.
Note that Q◦ = ∅, Q− = R, and ∂Q = R.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Interiors.
Every subset of a topological space contains an open set,namely the empty set.
Suppose A ⊂ (X , τ) is given and consider the nonemptycollection
{B ⊂ X | B ∈ τ, B ⊂ A} .
The union of this family’s members is an open set contained inA and is denoted by A◦.
Observe that A◦◦ = A◦, and if C ∈ τ , then C ◦ = C .
Given any A ⊂ (X , τ) the space X partitions as
X = A◦ ∪ ∂A ∪ Ac◦.
Note that Q◦ = ∅, Q− = R, and ∂Q = R.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Subspace topology.
Let (X , τ) be a topological space and A a subset of X .
The collection of subsets U ⊂ A such that U = A ∩ V , whereV ∈ τ , is in fact a topology in A.
This topology is know as the subspace topology.
It turns each subset into a topological space in its own right.
When there is a chain A ⊂ B ⊂ X , then write τB for thesubspace topology on B induced from X and similarly τA forthe one induced on A by X .
There is also a topology τBA on A induced by τB on B.
It is a technical exercise to verify that τA = τBA .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Subspace topology.
Let (X , τ) be a topological space and A a subset of X .
The collection of subsets U ⊂ A such that U = A ∩ V , whereV ∈ τ , is in fact a topology in A.
This topology is know as the subspace topology.
It turns each subset into a topological space in its own right.
When there is a chain A ⊂ B ⊂ X , then write τB for thesubspace topology on B induced from X and similarly τA forthe one induced on A by X .
There is also a topology τBA on A induced by τB on B.
It is a technical exercise to verify that τA = τBA .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Subspace topology.
Let (X , τ) be a topological space and A a subset of X .
The collection of subsets U ⊂ A such that U = A ∩ V , whereV ∈ τ , is in fact a topology in A.
This topology is know as the subspace topology.
It turns each subset into a topological space in its own right.
When there is a chain A ⊂ B ⊂ X , then write τB for thesubspace topology on B induced from X and similarly τA forthe one induced on A by X .
There is also a topology τBA on A induced by τB on B.
It is a technical exercise to verify that τA = τBA .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Subspace topology.
Let (X , τ) be a topological space and A a subset of X .
The collection of subsets U ⊂ A such that U = A ∩ V , whereV ∈ τ , is in fact a topology in A.
This topology is know as the subspace topology.
It turns each subset into a topological space in its own right.
When there is a chain A ⊂ B ⊂ X , then write τB for thesubspace topology on B induced from X and similarly τA forthe one induced on A by X .
There is also a topology τBA on A induced by τB on B.
It is a technical exercise to verify that τA = τBA .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Subspace topology.
Let (X , τ) be a topological space and A a subset of X .
The collection of subsets U ⊂ A such that U = A ∩ V , whereV ∈ τ , is in fact a topology in A.
This topology is know as the subspace topology.
It turns each subset into a topological space in its own right.
When there is a chain A ⊂ B ⊂ X , then write τB for thesubspace topology on B induced from X and similarly τA forthe one induced on A by X .
There is also a topology τBA on A induced by τB on B.
It is a technical exercise to verify that τA = τBA .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Subspace topology.
Let (X , τ) be a topological space and A a subset of X .
The collection of subsets U ⊂ A such that U = A ∩ V , whereV ∈ τ , is in fact a topology in A.
This topology is know as the subspace topology.
It turns each subset into a topological space in its own right.
When there is a chain A ⊂ B ⊂ X , then write τB for thesubspace topology on B induced from X and similarly τA forthe one induced on A by X .
There is also a topology τBA on A induced by τB on B.
It is a technical exercise to verify that τA = τBA .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Subspace topology.
Let (X , τ) be a topological space and A a subset of X .
The collection of subsets U ⊂ A such that U = A ∩ V , whereV ∈ τ , is in fact a topology in A.
This topology is know as the subspace topology.
It turns each subset into a topological space in its own right.
When there is a chain A ⊂ B ⊂ X , then write τB for thesubspace topology on B induced from X and similarly τA forthe one induced on A by X .
There is also a topology τBA on A induced by τB on B.
It is a technical exercise to verify that τA = τBA .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ (X , τ) is connected and dense in X , then X isconnected.
To see this, assume U is a nontrivial open and closed set in X .
It follows that U ∩ A and Uc ∩ A are open in the subspacetopology of A.
Since A is dense, neither of these two sets are empty.
Moreover, if U ∩ A = A, then A ⊂ U, which again isincompatible with A being dense.
Similarly, Uc ∩ A = A is also impossible.
It follows that U ∩ A is both open and closed in the subspacetopology of A, and hence A is not connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ (X , τ) is connected and dense in X , then X isconnected.
To see this, assume U is a nontrivial open and closed set in X .
It follows that U ∩ A and Uc ∩ A are open in the subspacetopology of A.
Since A is dense, neither of these two sets are empty.
Moreover, if U ∩ A = A, then A ⊂ U, which again isincompatible with A being dense.
Similarly, Uc ∩ A = A is also impossible.
It follows that U ∩ A is both open and closed in the subspacetopology of A, and hence A is not connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ (X , τ) is connected and dense in X , then X isconnected.
To see this, assume U is a nontrivial open and closed set in X .
It follows that U ∩ A and Uc ∩ A are open in the subspacetopology of A.
Since A is dense, neither of these two sets are empty.
Moreover, if U ∩ A = A, then A ⊂ U, which again isincompatible with A being dense.
Similarly, Uc ∩ A = A is also impossible.
It follows that U ∩ A is both open and closed in the subspacetopology of A, and hence A is not connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ (X , τ) is connected and dense in X , then X isconnected.
To see this, assume U is a nontrivial open and closed set in X .
It follows that U ∩ A and Uc ∩ A are open in the subspacetopology of A.
Since A is dense, neither of these two sets are empty.
Moreover, if U ∩ A = A, then A ⊂ U, which again isincompatible with A being dense.
Similarly, Uc ∩ A = A is also impossible.
It follows that U ∩ A is both open and closed in the subspacetopology of A, and hence A is not connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ (X , τ) is connected and dense in X , then X isconnected.
To see this, assume U is a nontrivial open and closed set in X .
It follows that U ∩ A and Uc ∩ A are open in the subspacetopology of A.
Since A is dense, neither of these two sets are empty.
Moreover, if U ∩ A = A, then A ⊂ U, which again isincompatible with A being dense.
Similarly, Uc ∩ A = A is also impossible.
It follows that U ∩ A is both open and closed in the subspacetopology of A, and hence A is not connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ (X , τ) is connected and dense in X , then X isconnected.
To see this, assume U is a nontrivial open and closed set in X .
It follows that U ∩ A and Uc ∩ A are open in the subspacetopology of A.
Since A is dense, neither of these two sets are empty.
Moreover, if U ∩ A = A, then A ⊂ U, which again isincompatible with A being dense.
Similarly, Uc ∩ A = A is also impossible.
It follows that U ∩ A is both open and closed in the subspacetopology of A, and hence A is not connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ (X , τ) is connected and dense in X , then X isconnected.
To see this, assume U is a nontrivial open and closed set in X .
It follows that U ∩ A and Uc ∩ A are open in the subspacetopology of A.
Since A is dense, neither of these two sets are empty.
Moreover, if U ∩ A = A, then A ⊂ U, which again isincompatible with A being dense.
Similarly, Uc ∩ A = A is also impossible.
It follows that U ∩ A is both open and closed in the subspacetopology of A, and hence A is not connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ B ⊂ A− and A is connected, then B is connected.
In particular if A is connected, then A− is connected.
The ‘closure operation’ does not ‘rip a space apart’.
Let (X , τ) be a topological space and write τ c for thecollection of closed sets.
Denote the collection of all subsets of X by P(X ).
Let φ : P(X )→ P(X ) be given φ(A) = Ac .
One may consider the restricted maps φ : τ → τ c andφ : τ c → τ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ B ⊂ A− and A is connected, then B is connected.
In particular if A is connected, then A− is connected.
The ‘closure operation’ does not ‘rip a space apart’.
Let (X , τ) be a topological space and write τ c for thecollection of closed sets.
Denote the collection of all subsets of X by P(X ).
Let φ : P(X )→ P(X ) be given φ(A) = Ac .
One may consider the restricted maps φ : τ → τ c andφ : τ c → τ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ B ⊂ A− and A is connected, then B is connected.
In particular if A is connected, then A− is connected.
The ‘closure operation’ does not ‘rip a space apart’.
Let (X , τ) be a topological space and write τ c for thecollection of closed sets.
Denote the collection of all subsets of X by P(X ).
Let φ : P(X )→ P(X ) be given φ(A) = Ac .
One may consider the restricted maps φ : τ → τ c andφ : τ c → τ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ B ⊂ A− and A is connected, then B is connected.
In particular if A is connected, then A− is connected.
The ‘closure operation’ does not ‘rip a space apart’.
Let (X , τ) be a topological space and write τ c for thecollection of closed sets.
Denote the collection of all subsets of X by P(X ).
Let φ : P(X )→ P(X ) be given φ(A) = Ac .
One may consider the restricted maps φ : τ → τ c andφ : τ c → τ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ B ⊂ A− and A is connected, then B is connected.
In particular if A is connected, then A− is connected.
The ‘closure operation’ does not ‘rip a space apart’.
Let (X , τ) be a topological space and write τ c for thecollection of closed sets.
Denote the collection of all subsets of X by P(X ).
Let φ : P(X )→ P(X ) be given φ(A) = Ac .
One may consider the restricted maps φ : τ → τ c andφ : τ c → τ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ B ⊂ A− and A is connected, then B is connected.
In particular if A is connected, then A− is connected.
The ‘closure operation’ does not ‘rip a space apart’.
Let (X , τ) be a topological space and write τ c for thecollection of closed sets.
Denote the collection of all subsets of X by P(X ).
Let φ : P(X )→ P(X ) be given φ(A) = Ac .
One may consider the restricted maps φ : τ → τ c andφ : τ c → τ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Connected sets and closures.
If A ⊂ B ⊂ A− and A is connected, then B is connected.
In particular if A is connected, then A− is connected.
The ‘closure operation’ does not ‘rip a space apart’.
Let (X , τ) be a topological space and write τ c for thecollection of closed sets.
Denote the collection of all subsets of X by P(X ).
Let φ : P(X )→ P(X ) be given φ(A) = Ac .
One may consider the restricted maps φ : τ → τ c andφ : τ c → τ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures and complements.
Consider the map ϕ : P(X )→ P(X ) given by ϕ(A) = A−.
Observe that ϕ(P(X )) = τ c .
It is also true that φ ◦ φ(A) = A and ϕ ◦ ϕ(A) = ϕ(A).
It is reasonable to ask what may be said about ϕ ◦ φ, φ ◦ ϕ, orφ ◦ ϕ ◦ φ, ϕ ◦ φ ◦ ϕ, and the analogous continuations.
Specifically, what may said about the following tree?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures and complements.
Consider the map ϕ : P(X )→ P(X ) given by ϕ(A) = A−.
Observe that ϕ(P(X )) = τ c .
It is also true that φ ◦ φ(A) = A and ϕ ◦ ϕ(A) = ϕ(A).
It is reasonable to ask what may be said about ϕ ◦ φ, φ ◦ ϕ, orφ ◦ ϕ ◦ φ, ϕ ◦ φ ◦ ϕ, and the analogous continuations.
Specifically, what may said about the following tree?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures and complements.
Consider the map ϕ : P(X )→ P(X ) given by ϕ(A) = A−.
Observe that ϕ(P(X )) = τ c .
It is also true that φ ◦ φ(A) = A and ϕ ◦ ϕ(A) = ϕ(A).
It is reasonable to ask what may be said about ϕ ◦ φ, φ ◦ ϕ, orφ ◦ ϕ ◦ φ, ϕ ◦ φ ◦ ϕ, and the analogous continuations.
Specifically, what may said about the following tree?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures and complements.
Consider the map ϕ : P(X )→ P(X ) given by ϕ(A) = A−.
Observe that ϕ(P(X )) = τ c .
It is also true that φ ◦ φ(A) = A and ϕ ◦ ϕ(A) = ϕ(A).
It is reasonable to ask what may be said about ϕ ◦ φ, φ ◦ ϕ, orφ ◦ ϕ ◦ φ, ϕ ◦ φ ◦ ϕ, and the analogous continuations.
Specifically, what may said about the following tree?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Closures and complements.
Consider the map ϕ : P(X )→ P(X ) given by ϕ(A) = A−.
Observe that ϕ(P(X )) = τ c .
It is also true that φ ◦ φ(A) = A and ϕ ◦ ϕ(A) = ϕ(A).
It is reasonable to ask what may be said about ϕ ◦ φ, φ ◦ ϕ, orφ ◦ ϕ ◦ φ, ϕ ◦ φ ◦ ϕ, and the analogous continuations.
Specifically, what may said about the following tree?
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A↙ ↘
Ac A−
↓ ↓Ac− A−c
↓ ↓Ac−c A−c−
↓ ↓...
...
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Whereas the concepts of separation and a connected spaceinvolve fairly immediate geometric ideas, the concept of acompact space does not.
The term compact may not capture the essence of theunderlying idea, and words like confined or inescapable woulddo a better job.
Suppose (X , τ) is a topological space, and let Uλ ∈ τ be acollection of subsets of X indexed by λ.
The collection is said to be an open cover of X if its union isall of X .
The space X is said to be compact if from any open cover it ispossible to extract a finite number of members that still forman open cover.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Whereas the concepts of separation and a connected spaceinvolve fairly immediate geometric ideas, the concept of acompact space does not.
The term compact may not capture the essence of theunderlying idea, and words like confined or inescapable woulddo a better job.
Suppose (X , τ) is a topological space, and let Uλ ∈ τ be acollection of subsets of X indexed by λ.
The collection is said to be an open cover of X if its union isall of X .
The space X is said to be compact if from any open cover it ispossible to extract a finite number of members that still forman open cover.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Whereas the concepts of separation and a connected spaceinvolve fairly immediate geometric ideas, the concept of acompact space does not.
The term compact may not capture the essence of theunderlying idea, and words like confined or inescapable woulddo a better job.
Suppose (X , τ) is a topological space, and let Uλ ∈ τ be acollection of subsets of X indexed by λ.
The collection is said to be an open cover of X if its union isall of X .
The space X is said to be compact if from any open cover it ispossible to extract a finite number of members that still forman open cover.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Whereas the concepts of separation and a connected spaceinvolve fairly immediate geometric ideas, the concept of acompact space does not.
The term compact may not capture the essence of theunderlying idea, and words like confined or inescapable woulddo a better job.
Suppose (X , τ) is a topological space, and let Uλ ∈ τ be acollection of subsets of X indexed by λ.
The collection is said to be an open cover of X if its union isall of X .
The space X is said to be compact if from any open cover it ispossible to extract a finite number of members that still forman open cover.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Whereas the concepts of separation and a connected spaceinvolve fairly immediate geometric ideas, the concept of acompact space does not.
The term compact may not capture the essence of theunderlying idea, and words like confined or inescapable woulddo a better job.
Suppose (X , τ) is a topological space, and let Uλ ∈ τ be acollection of subsets of X indexed by λ.
The collection is said to be an open cover of X if its union isall of X .
The space X is said to be compact if from any open cover it ispossible to extract a finite number of members that still forman open cover.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
The extracted open sets are known as a finite subcover of theoriginal open cover.
The collection {(
1
k, 1)
}∞k=2
is an open cover of (0, 1) that has no finite subcover.
Each finite topological space is compact since each open covermust be finite.
More generally, a topological space with only a finite number ofopen sets must be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
The extracted open sets are known as a finite subcover of theoriginal open cover.
The collection {(
1
k, 1)
}∞k=2
is an open cover of (0, 1) that has no finite subcover.
Each finite topological space is compact since each open covermust be finite.
More generally, a topological space with only a finite number ofopen sets must be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
The extracted open sets are known as a finite subcover of theoriginal open cover.
The collection {(
1
k, 1)
}∞k=2
is an open cover of (0, 1) that has no finite subcover.
Each finite topological space is compact since each open covermust be finite.
More generally, a topological space with only a finite number ofopen sets must be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
The extracted open sets are known as a finite subcover of theoriginal open cover.
The collection {(
1
k, 1)
}∞k=2
is an open cover of (0, 1) that has no finite subcover.
Each finite topological space is compact since each open covermust be finite.
More generally, a topological space with only a finite number ofopen sets must be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
Let Uλ be an open cover of [0, 1].
Let A be the set a ∈ [0, 1] such that [0, a] is contained in afinite number of the Uλ.
Since some Uλ contains 0, it must be that A 6= ∅.
It is also true that A is bounded above by 1.
Let α be the least upper bound of A.
If α < 1, then for some index λ0, α ∈ Uλ0 .
The Uλ0 that contains α may be added to A so that [0, a] iscovered by a finite number of open sets the open cover wherea > α, which is impossible.
Conclude that α = 1 is the only possibility.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
There is a Uλ1 in the original open cover such that 1 ∈ Uλ1 .Add Uλ1 to the finite open cover of A.
There is no x ∈ [0, 1) that is neither in Uλ1 nor in A becausesuch an x would be an upper bound less than α.
It follows that [0, 1] is compact.
In Rn it is well-known that a subset is compact if and only if itis closed and bounded.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
There is a Uλ1 in the original open cover such that 1 ∈ Uλ1 .Add Uλ1 to the finite open cover of A.
There is no x ∈ [0, 1) that is neither in Uλ1 nor in A becausesuch an x would be an upper bound less than α.
It follows that [0, 1] is compact.
In Rn it is well-known that a subset is compact if and only if itis closed and bounded.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
There is a Uλ1 in the original open cover such that 1 ∈ Uλ1 .Add Uλ1 to the finite open cover of A.
There is no x ∈ [0, 1) that is neither in Uλ1 nor in A becausesuch an x would be an upper bound less than α.
It follows that [0, 1] is compact.
In Rn it is well-known that a subset is compact if and only if itis closed and bounded.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
There is a Uλ1 in the original open cover such that 1 ∈ Uλ1 .Add Uλ1 to the finite open cover of A.
There is no x ∈ [0, 1) that is neither in Uλ1 nor in A becausesuch an x would be an upper bound less than α.
It follows that [0, 1] is compact.
In Rn it is well-known that a subset is compact if and only if itis closed and bounded.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
When a subset is regarded as a topological space in its ownright, using the subspace topology, it is automatically closed.
The issue that comes up in general is whether when A ⊂ X iscompact it is also closed as a subset in X .
Since every non-trivial subset of a space with the indiscretetopology is compact and none is closed, the answer in generalis no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
When a subset is regarded as a topological space in its ownright, using the subspace topology, it is automatically closed.
The issue that comes up in general is whether when A ⊂ X iscompact it is also closed as a subset in X .
Since every non-trivial subset of a space with the indiscretetopology is compact and none is closed, the answer in generalis no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compact spaces.
When a subset is regarded as a topological space in its ownright, using the subspace topology, it is automatically closed.
The issue that comes up in general is whether when A ⊂ X iscompact it is also closed as a subset in X .
Since every non-trivial subset of a space with the indiscretetopology is compact and none is closed, the answer in generalis no.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Let (X0, τ0) and (X1, τ1) be two topological spaces.
A function f : X0 → X1 is said to be open if f (A) ∈ τ1
whenever A ∈ τ0.
A function f : X0 → X1 is said to be closed if f (A) ∈ τ c1whenever A ∈ τ c0 .
A function f : X0 → X1 is said to be continuous if f −1(A) ∈ τ0
whenever A ∈ τ1.
In this last case it is not at all assumed that f is invertible.
If f is invertible, and both f and f −1 are open, then f is ahomeomorphism,
This amounts to the fact that both f and f −1 are continuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Let (X0, τ0) and (X1, τ1) be two topological spaces.
A function f : X0 → X1 is said to be open if f (A) ∈ τ1
whenever A ∈ τ0.
A function f : X0 → X1 is said to be closed if f (A) ∈ τ c1whenever A ∈ τ c0 .
A function f : X0 → X1 is said to be continuous if f −1(A) ∈ τ0
whenever A ∈ τ1.
In this last case it is not at all assumed that f is invertible.
If f is invertible, and both f and f −1 are open, then f is ahomeomorphism,
This amounts to the fact that both f and f −1 are continuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Let (X0, τ0) and (X1, τ1) be two topological spaces.
A function f : X0 → X1 is said to be open if f (A) ∈ τ1
whenever A ∈ τ0.
A function f : X0 → X1 is said to be closed if f (A) ∈ τ c1whenever A ∈ τ c0 .
A function f : X0 → X1 is said to be continuous if f −1(A) ∈ τ0
whenever A ∈ τ1.
In this last case it is not at all assumed that f is invertible.
If f is invertible, and both f and f −1 are open, then f is ahomeomorphism,
This amounts to the fact that both f and f −1 are continuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Let (X0, τ0) and (X1, τ1) be two topological spaces.
A function f : X0 → X1 is said to be open if f (A) ∈ τ1
whenever A ∈ τ0.
A function f : X0 → X1 is said to be closed if f (A) ∈ τ c1whenever A ∈ τ c0 .
A function f : X0 → X1 is said to be continuous if f −1(A) ∈ τ0
whenever A ∈ τ1.
In this last case it is not at all assumed that f is invertible.
If f is invertible, and both f and f −1 are open, then f is ahomeomorphism,
This amounts to the fact that both f and f −1 are continuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Let (X0, τ0) and (X1, τ1) be two topological spaces.
A function f : X0 → X1 is said to be open if f (A) ∈ τ1
whenever A ∈ τ0.
A function f : X0 → X1 is said to be closed if f (A) ∈ τ c1whenever A ∈ τ c0 .
A function f : X0 → X1 is said to be continuous if f −1(A) ∈ τ0
whenever A ∈ τ1.
In this last case it is not at all assumed that f is invertible.
If f is invertible, and both f and f −1 are open, then f is ahomeomorphism,
This amounts to the fact that both f and f −1 are continuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Let (X0, τ0) and (X1, τ1) be two topological spaces.
A function f : X0 → X1 is said to be open if f (A) ∈ τ1
whenever A ∈ τ0.
A function f : X0 → X1 is said to be closed if f (A) ∈ τ c1whenever A ∈ τ c0 .
A function f : X0 → X1 is said to be continuous if f −1(A) ∈ τ0
whenever A ∈ τ1.
In this last case it is not at all assumed that f is invertible.
If f is invertible, and both f and f −1 are open, then f is ahomeomorphism,
This amounts to the fact that both f and f −1 are continuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Let (X0, τ0) and (X1, τ1) be two topological spaces.
A function f : X0 → X1 is said to be open if f (A) ∈ τ1
whenever A ∈ τ0.
A function f : X0 → X1 is said to be closed if f (A) ∈ τ c1whenever A ∈ τ c0 .
A function f : X0 → X1 is said to be continuous if f −1(A) ∈ τ0
whenever A ∈ τ1.
In this last case it is not at all assumed that f is invertible.
If f is invertible, and both f and f −1 are open, then f is ahomeomorphism,
This amounts to the fact that both f and f −1 are continuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Assume f : X0 → X1 is continuous, so it is sometimes called amap.
If A is connected, then f (A) is connected.
If A is compact, then f (A) is compact.
If X is a topological space and γ : [0, 1]→ X is a map, then γis said to be a path in X .
The point γ(0) is called the initial point, and γ(1) the finalpoint of γ.
If to each pair p, q ∈ X there is a path γp,q such thatγp,q(0) = p and γp,q(1) = q, then X is said to bepath-connected.
Path-connected spaces are connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Assume f : X0 → X1 is continuous, so it is sometimes called amap.
If A is connected, then f (A) is connected.
If A is compact, then f (A) is compact.
If X is a topological space and γ : [0, 1]→ X is a map, then γis said to be a path in X .
The point γ(0) is called the initial point, and γ(1) the finalpoint of γ.
If to each pair p, q ∈ X there is a path γp,q such thatγp,q(0) = p and γp,q(1) = q, then X is said to bepath-connected.
Path-connected spaces are connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Assume f : X0 → X1 is continuous, so it is sometimes called amap.
If A is connected, then f (A) is connected.
If A is compact, then f (A) is compact.
If X is a topological space and γ : [0, 1]→ X is a map, then γis said to be a path in X .
The point γ(0) is called the initial point, and γ(1) the finalpoint of γ.
If to each pair p, q ∈ X there is a path γp,q such thatγp,q(0) = p and γp,q(1) = q, then X is said to bepath-connected.
Path-connected spaces are connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Assume f : X0 → X1 is continuous, so it is sometimes called amap.
If A is connected, then f (A) is connected.
If A is compact, then f (A) is compact.
If X is a topological space and γ : [0, 1]→ X is a map, then γis said to be a path in X .
The point γ(0) is called the initial point, and γ(1) the finalpoint of γ.
If to each pair p, q ∈ X there is a path γp,q such thatγp,q(0) = p and γp,q(1) = q, then X is said to bepath-connected.
Path-connected spaces are connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Assume f : X0 → X1 is continuous, so it is sometimes called amap.
If A is connected, then f (A) is connected.
If A is compact, then f (A) is compact.
If X is a topological space and γ : [0, 1]→ X is a map, then γis said to be a path in X .
The point γ(0) is called the initial point, and γ(1) the finalpoint of γ.
If to each pair p, q ∈ X there is a path γp,q such thatγp,q(0) = p and γp,q(1) = q, then X is said to bepath-connected.
Path-connected spaces are connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Assume f : X0 → X1 is continuous, so it is sometimes called amap.
If A is connected, then f (A) is connected.
If A is compact, then f (A) is compact.
If X is a topological space and γ : [0, 1]→ X is a map, then γis said to be a path in X .
The point γ(0) is called the initial point, and γ(1) the finalpoint of γ.
If to each pair p, q ∈ X there is a path γp,q such thatγp,q(0) = p and γp,q(1) = q, then X is said to bepath-connected.
Path-connected spaces are connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Maps.
Assume f : X0 → X1 is continuous, so it is sometimes called amap.
If A is connected, then f (A) is connected.
If A is compact, then f (A) is compact.
If X is a topological space and γ : [0, 1]→ X is a map, then γis said to be a path in X .
The point γ(0) is called the initial point, and γ(1) the finalpoint of γ.
If to each pair p, q ∈ X there is a path γp,q such thatγp,q(0) = p and γp,q(1) = q, then X is said to bepath-connected.
Path-connected spaces are connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and X compact.
Let Vλ be an open cover of f (X ).
If follows that Uλ = f −1(Vλ) is an open cover of X .
Since X is compact, there is a finite open subcover Uλk wherek ∈ {1, . . . ,K}.
Consider the subcollection Vλk .
It is in fact a finite subcover.
To see this, let y ∈ f (X ) so that y = f (x).
There is a k such that x ∈ Uλk .
It follows that y ∈ Vλk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and f (X ) not connected.
Suppose V ⊂ f (X ) is nontrivial open and closed set.
Let U = f −1(V ) and observe that U is nontrivial, and alsoopen by the continuity of f .
Since V is closed, it follows that f (X )− V is open.
There is some open set W with W ⊂ Y such thatf (X )− V = f (X ) ∩W .
Now f −1(W ) is open in X .
The claim is that X − U = f −1(W ).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and f (X ) not connected.
Suppose V ⊂ f (X ) is nontrivial open and closed set.
Let U = f −1(V ) and observe that U is nontrivial, and alsoopen by the continuity of f .
Since V is closed, it follows that f (X )− V is open.
There is some open set W with W ⊂ Y such thatf (X )− V = f (X ) ∩W .
Now f −1(W ) is open in X .
The claim is that X − U = f −1(W ).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and f (X ) not connected.
Suppose V ⊂ f (X ) is nontrivial open and closed set.
Let U = f −1(V ) and observe that U is nontrivial, and alsoopen by the continuity of f .
Since V is closed, it follows that f (X )− V is open.
There is some open set W with W ⊂ Y such thatf (X )− V = f (X ) ∩W .
Now f −1(W ) is open in X .
The claim is that X − U = f −1(W ).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and f (X ) not connected.
Suppose V ⊂ f (X ) is nontrivial open and closed set.
Let U = f −1(V ) and observe that U is nontrivial, and alsoopen by the continuity of f .
Since V is closed, it follows that f (X )− V is open.
There is some open set W with W ⊂ Y such thatf (X )− V = f (X ) ∩W .
Now f −1(W ) is open in X .
The claim is that X − U = f −1(W ).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and f (X ) not connected.
Suppose V ⊂ f (X ) is nontrivial open and closed set.
Let U = f −1(V ) and observe that U is nontrivial, and alsoopen by the continuity of f .
Since V is closed, it follows that f (X )− V is open.
There is some open set W with W ⊂ Y such thatf (X )− V = f (X ) ∩W .
Now f −1(W ) is open in X .
The claim is that X − U = f −1(W ).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and f (X ) not connected.
Suppose V ⊂ f (X ) is nontrivial open and closed set.
Let U = f −1(V ) and observe that U is nontrivial, and alsoopen by the continuity of f .
Since V is closed, it follows that f (X )− V is open.
There is some open set W with W ⊂ Y such thatf (X )− V = f (X ) ∩W .
Now f −1(W ) is open in X .
The claim is that X − U = f −1(W ).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and f (X ) not connected.
Suppose V ⊂ f (X ) is nontrivial open and closed set.
Let U = f −1(V ) and observe that U is nontrivial, and alsoopen by the continuity of f .
Since V is closed, it follows that f (X )− V is open.
There is some open set W with W ⊂ Y such thatf (X )− V = f (X ) ∩W .
Now f −1(W ) is open in X .
The claim is that X − U = f −1(W ).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose x ∈ X − U and observe that f (x) ∈ f (X )− V sof (x) ∈W .
Conversely suppose x ∈ f −1(W ) so that f (x) ∈W .
It follows that f (x) /∈ V and hence x /∈ U.
One implication of this result is seen in Calculus and theintermediate value theorem.
Observe that the image of a path is always both connected andcompact.
There are disturbing examples of so-called space-filling curveswhere, for instance, f ([0, 1]) = [0, 1]× [0, 1].
Knowing such an example proves that [0, 1]× [0, 1] is compact,which is a nontrivial result.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose x ∈ X − U and observe that f (x) ∈ f (X )− V sof (x) ∈W .
Conversely suppose x ∈ f −1(W ) so that f (x) ∈W .
It follows that f (x) /∈ V and hence x /∈ U.
One implication of this result is seen in Calculus and theintermediate value theorem.
Observe that the image of a path is always both connected andcompact.
There are disturbing examples of so-called space-filling curveswhere, for instance, f ([0, 1]) = [0, 1]× [0, 1].
Knowing such an example proves that [0, 1]× [0, 1] is compact,which is a nontrivial result.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose x ∈ X − U and observe that f (x) ∈ f (X )− V sof (x) ∈W .
Conversely suppose x ∈ f −1(W ) so that f (x) ∈W .
It follows that f (x) /∈ V and hence x /∈ U.
One implication of this result is seen in Calculus and theintermediate value theorem.
Observe that the image of a path is always both connected andcompact.
There are disturbing examples of so-called space-filling curveswhere, for instance, f ([0, 1]) = [0, 1]× [0, 1].
Knowing such an example proves that [0, 1]× [0, 1] is compact,which is a nontrivial result.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose x ∈ X − U and observe that f (x) ∈ f (X )− V sof (x) ∈W .
Conversely suppose x ∈ f −1(W ) so that f (x) ∈W .
It follows that f (x) /∈ V and hence x /∈ U.
One implication of this result is seen in Calculus and theintermediate value theorem.
Observe that the image of a path is always both connected andcompact.
There are disturbing examples of so-called space-filling curveswhere, for instance, f ([0, 1]) = [0, 1]× [0, 1].
Knowing such an example proves that [0, 1]× [0, 1] is compact,which is a nontrivial result.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose x ∈ X − U and observe that f (x) ∈ f (X )− V sof (x) ∈W .
Conversely suppose x ∈ f −1(W ) so that f (x) ∈W .
It follows that f (x) /∈ V and hence x /∈ U.
One implication of this result is seen in Calculus and theintermediate value theorem.
Observe that the image of a path is always both connected andcompact.
There are disturbing examples of so-called space-filling curveswhere, for instance, f ([0, 1]) = [0, 1]× [0, 1].
Knowing such an example proves that [0, 1]× [0, 1] is compact,which is a nontrivial result.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose x ∈ X − U and observe that f (x) ∈ f (X )− V sof (x) ∈W .
Conversely suppose x ∈ f −1(W ) so that f (x) ∈W .
It follows that f (x) /∈ V and hence x /∈ U.
One implication of this result is seen in Calculus and theintermediate value theorem.
Observe that the image of a path is always both connected andcompact.
There are disturbing examples of so-called space-filling curveswhere, for instance, f ([0, 1]) = [0, 1]× [0, 1].
Knowing such an example proves that [0, 1]× [0, 1] is compact,which is a nontrivial result.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose x ∈ X − U and observe that f (x) ∈ f (X )− V sof (x) ∈W .
Conversely suppose x ∈ f −1(W ) so that f (x) ∈W .
It follows that f (x) /∈ V and hence x /∈ U.
One implication of this result is seen in Calculus and theintermediate value theorem.
Observe that the image of a path is always both connected andcompact.
There are disturbing examples of so-called space-filling curveswhere, for instance, f ([0, 1]) = [0, 1]× [0, 1].
Knowing such an example proves that [0, 1]× [0, 1] is compact,which is a nontrivial result.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The space
Z ={
(t, sin(π
t)) | t ∈ (0, 1]
}⊂ R2
is path-connected, and hence connected.
Let Y = {(0, y) | −1 ≤ y ≤ 1} and observe thatX = Z− = Y ∪ Z is connected.
It is very demanding technically to prove that X is notpath-connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The space
Z ={
(t, sin(π
t)) | t ∈ (0, 1]
}⊂ R2
is path-connected, and hence connected.
Let Y = {(0, y) | −1 ≤ y ≤ 1} and observe thatX = Z− = Y ∪ Z is connected.
It is very demanding technically to prove that X is notpath-connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The space
Z ={
(t, sin(π
t)) | t ∈ (0, 1]
}⊂ R2
is path-connected, and hence connected.
Let Y = {(0, y) | −1 ≤ y ≤ 1} and observe thatX = Z− = Y ∪ Z is connected.
It is very demanding technically to prove that X is notpath-connected.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is a topological space and f : X → R a givenfunction.
Suppose f is bounded from below so that the greatest lowerbound α ∈ R of f (X ) exists.
One of the most important tasks in all of mathematics is toestablish conditions on X and f that guarantee that there issome x ∈ X such that f (x) = α.
There is of course a sequence of points xk ∈ X such that f (xk)tends to α.
The example f : R→ R given by f (x) = ex shows thatxk = −k has f (xk) tending to α = 0, whereas the xk do nottend to any real number.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is a topological space and f : X → R a givenfunction.
Suppose f is bounded from below so that the greatest lowerbound α ∈ R of f (X ) exists.
One of the most important tasks in all of mathematics is toestablish conditions on X and f that guarantee that there issome x ∈ X such that f (x) = α.
There is of course a sequence of points xk ∈ X such that f (xk)tends to α.
The example f : R→ R given by f (x) = ex shows thatxk = −k has f (xk) tending to α = 0, whereas the xk do nottend to any real number.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is a topological space and f : X → R a givenfunction.
Suppose f is bounded from below so that the greatest lowerbound α ∈ R of f (X ) exists.
One of the most important tasks in all of mathematics is toestablish conditions on X and f that guarantee that there issome x ∈ X such that f (x) = α.
There is of course a sequence of points xk ∈ X such that f (xk)tends to α.
The example f : R→ R given by f (x) = ex shows thatxk = −k has f (xk) tending to α = 0, whereas the xk do nottend to any real number.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is a topological space and f : X → R a givenfunction.
Suppose f is bounded from below so that the greatest lowerbound α ∈ R of f (X ) exists.
One of the most important tasks in all of mathematics is toestablish conditions on X and f that guarantee that there issome x ∈ X such that f (x) = α.
There is of course a sequence of points xk ∈ X such that f (xk)tends to α.
The example f : R→ R given by f (x) = ex shows thatxk = −k has f (xk) tending to α = 0, whereas the xk do nottend to any real number.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is a topological space and f : X → R a givenfunction.
Suppose f is bounded from below so that the greatest lowerbound α ∈ R of f (X ) exists.
One of the most important tasks in all of mathematics is toestablish conditions on X and f that guarantee that there issome x ∈ X such that f (x) = α.
There is of course a sequence of points xk ∈ X such that f (xk)tends to α.
The example f : R→ R given by f (x) = ex shows thatxk = −k has f (xk) tending to α = 0, whereas the xk do nottend to any real number.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A preorder � is reflexive, x � x , and transitive, x � y andy � z implies x � z .
A directed set D is a set with a preorder � such that to eachpair x , y ∈ D there is some z ∈ D such that x � z and y � z .
(N,≤) is a basic example of a directed set with �=≤.
Set-inclusion, �=⊂, and reverse set-inclusion, �=⊃, offer twomore useful examples.
A function x with domain a directed D is known as a net.
The most basic example is given by x : N→ X .
In this case x is called a sequence in X , and one writesx(k) = xk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A preorder � is reflexive, x � x , and transitive, x � y andy � z implies x � z .
A directed set D is a set with a preorder � such that to eachpair x , y ∈ D there is some z ∈ D such that x � z and y � z .
(N,≤) is a basic example of a directed set with �=≤.
Set-inclusion, �=⊂, and reverse set-inclusion, �=⊃, offer twomore useful examples.
A function x with domain a directed D is known as a net.
The most basic example is given by x : N→ X .
In this case x is called a sequence in X , and one writesx(k) = xk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A preorder � is reflexive, x � x , and transitive, x � y andy � z implies x � z .
A directed set D is a set with a preorder � such that to eachpair x , y ∈ D there is some z ∈ D such that x � z and y � z .
(N,≤) is a basic example of a directed set with �=≤.
Set-inclusion, �=⊂, and reverse set-inclusion, �=⊃, offer twomore useful examples.
A function x with domain a directed D is known as a net.
The most basic example is given by x : N→ X .
In this case x is called a sequence in X , and one writesx(k) = xk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A preorder � is reflexive, x � x , and transitive, x � y andy � z implies x � z .
A directed set D is a set with a preorder � such that to eachpair x , y ∈ D there is some z ∈ D such that x � z and y � z .
(N,≤) is a basic example of a directed set with �=≤.
Set-inclusion, �=⊂, and reverse set-inclusion, �=⊃, offer twomore useful examples.
A function x with domain a directed D is known as a net.
The most basic example is given by x : N→ X .
In this case x is called a sequence in X , and one writesx(k) = xk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A preorder � is reflexive, x � x , and transitive, x � y andy � z implies x � z .
A directed set D is a set with a preorder � such that to eachpair x , y ∈ D there is some z ∈ D such that x � z and y � z .
(N,≤) is a basic example of a directed set with �=≤.
Set-inclusion, �=⊂, and reverse set-inclusion, �=⊃, offer twomore useful examples.
A function x with domain a directed D is known as a net.
The most basic example is given by x : N→ X .
In this case x is called a sequence in X , and one writesx(k) = xk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A preorder � is reflexive, x � x , and transitive, x � y andy � z implies x � z .
A directed set D is a set with a preorder � such that to eachpair x , y ∈ D there is some z ∈ D such that x � z and y � z .
(N,≤) is a basic example of a directed set with �=≤.
Set-inclusion, �=⊂, and reverse set-inclusion, �=⊃, offer twomore useful examples.
A function x with domain a directed D is known as a net.
The most basic example is given by x : N→ X .
In this case x is called a sequence in X , and one writesx(k) = xk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A preorder � is reflexive, x � x , and transitive, x � y andy � z implies x � z .
A directed set D is a set with a preorder � such that to eachpair x , y ∈ D there is some z ∈ D such that x � z and y � z .
(N,≤) is a basic example of a directed set with �=≤.
Set-inclusion, �=⊂, and reverse set-inclusion, �=⊃, offer twomore useful examples.
A function x with domain a directed D is known as a net.
The most basic example is given by x : N→ X .
In this case x is called a sequence in X , and one writesx(k) = xk .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Assume x : D → X , where D is a directed set and (X , τ) is atopological space.
Write x(λ) = xλ ∈ X .
The net xλ is eventually in U ∈ τ if there is some λ0 such thatλ0 � λ implies xλ ∈ U.
If to each λ0 ∈ D there is a different λ1 ∈ D such that λ0 � λ1
and xλ1 ∈ U, then the net xλ is said to be frequently in U.
The net xλ converges to x∞ ∈ X if xλ is eventually in eachU ∈ τ with x∞ ∈ U.
The point x∞ is said to be a limit of the net xλ.
In Hausdorff spaces a limit is unique.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Assume x : D → X , where D is a directed set and (X , τ) is atopological space.
Write x(λ) = xλ ∈ X .
The net xλ is eventually in U ∈ τ if there is some λ0 such thatλ0 � λ implies xλ ∈ U.
If to each λ0 ∈ D there is a different λ1 ∈ D such that λ0 � λ1
and xλ1 ∈ U, then the net xλ is said to be frequently in U.
The net xλ converges to x∞ ∈ X if xλ is eventually in eachU ∈ τ with x∞ ∈ U.
The point x∞ is said to be a limit of the net xλ.
In Hausdorff spaces a limit is unique.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Assume x : D → X , where D is a directed set and (X , τ) is atopological space.
Write x(λ) = xλ ∈ X .
The net xλ is eventually in U ∈ τ if there is some λ0 such thatλ0 � λ implies xλ ∈ U.
If to each λ0 ∈ D there is a different λ1 ∈ D such that λ0 � λ1
and xλ1 ∈ U, then the net xλ is said to be frequently in U.
The net xλ converges to x∞ ∈ X if xλ is eventually in eachU ∈ τ with x∞ ∈ U.
The point x∞ is said to be a limit of the net xλ.
In Hausdorff spaces a limit is unique.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Assume x : D → X , where D is a directed set and (X , τ) is atopological space.
Write x(λ) = xλ ∈ X .
The net xλ is eventually in U ∈ τ if there is some λ0 such thatλ0 � λ implies xλ ∈ U.
If to each λ0 ∈ D there is a different λ1 ∈ D such that λ0 � λ1
and xλ1 ∈ U, then the net xλ is said to be frequently in U.
The net xλ converges to x∞ ∈ X if xλ is eventually in eachU ∈ τ with x∞ ∈ U.
The point x∞ is said to be a limit of the net xλ.
In Hausdorff spaces a limit is unique.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Assume x : D → X , where D is a directed set and (X , τ) is atopological space.
Write x(λ) = xλ ∈ X .
The net xλ is eventually in U ∈ τ if there is some λ0 such thatλ0 � λ implies xλ ∈ U.
If to each λ0 ∈ D there is a different λ1 ∈ D such that λ0 � λ1
and xλ1 ∈ U, then the net xλ is said to be frequently in U.
The net xλ converges to x∞ ∈ X if xλ is eventually in eachU ∈ τ with x∞ ∈ U.
The point x∞ is said to be a limit of the net xλ.
In Hausdorff spaces a limit is unique.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Assume x : D → X , where D is a directed set and (X , τ) is atopological space.
Write x(λ) = xλ ∈ X .
The net xλ is eventually in U ∈ τ if there is some λ0 such thatλ0 � λ implies xλ ∈ U.
If to each λ0 ∈ D there is a different λ1 ∈ D such that λ0 � λ1
and xλ1 ∈ U, then the net xλ is said to be frequently in U.
The net xλ converges to x∞ ∈ X if xλ is eventually in eachU ∈ τ with x∞ ∈ U.
The point x∞ is said to be a limit of the net xλ.
In Hausdorff spaces a limit is unique.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Assume x : D → X , where D is a directed set and (X , τ) is atopological space.
Write x(λ) = xλ ∈ X .
The net xλ is eventually in U ∈ τ if there is some λ0 such thatλ0 � λ implies xλ ∈ U.
If to each λ0 ∈ D there is a different λ1 ∈ D such that λ0 � λ1
and xλ1 ∈ U, then the net xλ is said to be frequently in U.
The net xλ converges to x∞ ∈ X if xλ is eventually in eachU ∈ τ with x∞ ∈ U.
The point x∞ is said to be a limit of the net xλ.
In Hausdorff spaces a limit is unique.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In any topological space (X , τ), U ⊂ X is open if and only ifno net in X − U has a limit in U.
Proof: Given U ∈ τ let xλ ∈ X − U.
Now it cannot be that xλ → x∞ ∈ U because U is open and xλis not in U.
Conversely, suppose A ⊂ X is not open and select x∞ ∈ Acarefully so that each U ∈ τ with x∞ ∈ U has U ∩ Ac 6= ∅.
This produces a net xU where the directed set is ordered byreverse inclusion.
Observe that if U � V then V ⊂ U and xV ∈ U so xU → x∞holds.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In any topological space (X , τ), U ⊂ X is open if and only ifno net in X − U has a limit in U.
Proof: Given U ∈ τ let xλ ∈ X − U.
Now it cannot be that xλ → x∞ ∈ U because U is open and xλis not in U.
Conversely, suppose A ⊂ X is not open and select x∞ ∈ Acarefully so that each U ∈ τ with x∞ ∈ U has U ∩ Ac 6= ∅.
This produces a net xU where the directed set is ordered byreverse inclusion.
Observe that if U � V then V ⊂ U and xV ∈ U so xU → x∞holds.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In any topological space (X , τ), U ⊂ X is open if and only ifno net in X − U has a limit in U.
Proof: Given U ∈ τ let xλ ∈ X − U.
Now it cannot be that xλ → x∞ ∈ U because U is open and xλis not in U.
Conversely, suppose A ⊂ X is not open and select x∞ ∈ Acarefully so that each U ∈ τ with x∞ ∈ U has U ∩ Ac 6= ∅.
This produces a net xU where the directed set is ordered byreverse inclusion.
Observe that if U � V then V ⊂ U and xV ∈ U so xU → x∞holds.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In any topological space (X , τ), U ⊂ X is open if and only ifno net in X − U has a limit in U.
Proof: Given U ∈ τ let xλ ∈ X − U.
Now it cannot be that xλ → x∞ ∈ U because U is open and xλis not in U.
Conversely, suppose A ⊂ X is not open and select x∞ ∈ Acarefully so that each U ∈ τ with x∞ ∈ U has U ∩ Ac 6= ∅.
This produces a net xU where the directed set is ordered byreverse inclusion.
Observe that if U � V then V ⊂ U and xV ∈ U so xU → x∞holds.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In any topological space (X , τ), U ⊂ X is open if and only ifno net in X − U has a limit in U.
Proof: Given U ∈ τ let xλ ∈ X − U.
Now it cannot be that xλ → x∞ ∈ U because U is open and xλis not in U.
Conversely, suppose A ⊂ X is not open and select x∞ ∈ Acarefully so that each U ∈ τ with x∞ ∈ U has U ∩ Ac 6= ∅.
This produces a net xU where the directed set is ordered byreverse inclusion.
Observe that if U � V then V ⊂ U and xV ∈ U so xU → x∞holds.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In any topological space (X , τ), U ⊂ X is open if and only ifno net in X − U has a limit in U.
Proof: Given U ∈ τ let xλ ∈ X − U.
Now it cannot be that xλ → x∞ ∈ U because U is open and xλis not in U.
Conversely, suppose A ⊂ X is not open and select x∞ ∈ Acarefully so that each U ∈ τ with x∞ ∈ U has U ∩ Ac 6= ∅.
This produces a net xU where the directed set is ordered byreverse inclusion.
Observe that if U � V then V ⊂ U and xV ∈ U so xU → x∞holds.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose A ⊂ X , then x∞ ∈ A− if and only if there is a netxλ ∈ A such that xλ → x∞ ∈ X .
Proof: Suppose xλ ∈ A and xλ → x∞ ∈ (X , τ).
Let U be open with x∞ ∈ U. There is a λU such that λU � λthen xλ ∈ U.
In particular, xλU ∈ U.
It follows that for each U ∈ τ with x∞ ∈ U it must be thatU ∩ A 6= ∅, and hence x∞ ∈ A−.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose A ⊂ X , then x∞ ∈ A− if and only if there is a netxλ ∈ A such that xλ → x∞ ∈ X .
Proof: Suppose xλ ∈ A and xλ → x∞ ∈ (X , τ).
Let U be open with x∞ ∈ U. There is a λU such that λU � λthen xλ ∈ U.
In particular, xλU ∈ U.
It follows that for each U ∈ τ with x∞ ∈ U it must be thatU ∩ A 6= ∅, and hence x∞ ∈ A−.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose A ⊂ X , then x∞ ∈ A− if and only if there is a netxλ ∈ A such that xλ → x∞ ∈ X .
Proof: Suppose xλ ∈ A and xλ → x∞ ∈ (X , τ).
Let U be open with x∞ ∈ U. There is a λU such that λU � λthen xλ ∈ U.
In particular, xλU ∈ U.
It follows that for each U ∈ τ with x∞ ∈ U it must be thatU ∩ A 6= ∅, and hence x∞ ∈ A−.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose A ⊂ X , then x∞ ∈ A− if and only if there is a netxλ ∈ A such that xλ → x∞ ∈ X .
Proof: Suppose xλ ∈ A and xλ → x∞ ∈ (X , τ).
Let U be open with x∞ ∈ U. There is a λU such that λU � λthen xλ ∈ U.
In particular, xλU ∈ U.
It follows that for each U ∈ τ with x∞ ∈ U it must be thatU ∩ A 6= ∅, and hence x∞ ∈ A−.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose A ⊂ X , then x∞ ∈ A− if and only if there is a netxλ ∈ A such that xλ → x∞ ∈ X .
Proof: Suppose xλ ∈ A and xλ → x∞ ∈ (X , τ).
Let U be open with x∞ ∈ U. There is a λU such that λU � λthen xλ ∈ U.
In particular, xλU ∈ U.
It follows that for each U ∈ τ with x∞ ∈ U it must be thatU ∩ A 6= ∅, and hence x∞ ∈ A−.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Conversely, suppose x∞ ∈ A−.
Consider the collection of U ∈ τ such that x∞ ∈ U.
Let � be defined by reverse inclusion so that a directed set isproduced.
For each U in the collection it is true that there is somexU ∈ A ∩ U since x∞ ∈ A−.
This produces a net xU in A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Conversely, suppose x∞ ∈ A−.
Consider the collection of U ∈ τ such that x∞ ∈ U.
Let � be defined by reverse inclusion so that a directed set isproduced.
For each U in the collection it is true that there is somexU ∈ A ∩ U since x∞ ∈ A−.
This produces a net xU in A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Conversely, suppose x∞ ∈ A−.
Consider the collection of U ∈ τ such that x∞ ∈ U.
Let � be defined by reverse inclusion so that a directed set isproduced.
For each U in the collection it is true that there is somexU ∈ A ∩ U since x∞ ∈ A−.
This produces a net xU in A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Conversely, suppose x∞ ∈ A−.
Consider the collection of U ∈ τ such that x∞ ∈ U.
Let � be defined by reverse inclusion so that a directed set isproduced.
For each U in the collection it is true that there is somexU ∈ A ∩ U since x∞ ∈ A−.
This produces a net xU in A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Conversely, suppose x∞ ∈ A−.
Consider the collection of U ∈ τ such that x∞ ∈ U.
Let � be defined by reverse inclusion so that a directed set isproduced.
For each U in the collection it is true that there is somexU ∈ A ∩ U since x∞ ∈ A−.
This produces a net xU in A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The claim now is that xU → x∞.
Suppose V ∈ τ has x∞ ∈ V .
Now if V � U then U ⊂ V and xU ∈ V .
This means xU → x∞.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The claim now is that xU → x∞.
Suppose V ∈ τ has x∞ ∈ V .
Now if V � U then U ⊂ V and xU ∈ V .
This means xU → x∞.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The claim now is that xU → x∞.
Suppose V ∈ τ has x∞ ∈ V .
Now if V � U then U ⊂ V and xU ∈ V .
This means xU → x∞.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The claim now is that xU → x∞.
Suppose V ∈ τ has x∞ ∈ V .
Now if V � U then U ⊂ V and xU ∈ V .
This means xU → x∞.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y , then f is continuous if and only if eachconvergent net xλ → x∞ satisfies f (xλ)→ f (x∞).
Proof: Suppose f : X → Y is continuous and let xλ → x∞ inX .
Consider the image f (x∞) ∈ Y .
Let V ∈ τY be such that f (x∞) ∈ V .
By continuity, U = f −1(V ) is open in X .
Observe that x∞ belongs to U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y , then f is continuous if and only if eachconvergent net xλ → x∞ satisfies f (xλ)→ f (x∞).
Proof: Suppose f : X → Y is continuous and let xλ → x∞ inX .
Consider the image f (x∞) ∈ Y .
Let V ∈ τY be such that f (x∞) ∈ V .
By continuity, U = f −1(V ) is open in X .
Observe that x∞ belongs to U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y , then f is continuous if and only if eachconvergent net xλ → x∞ satisfies f (xλ)→ f (x∞).
Proof: Suppose f : X → Y is continuous and let xλ → x∞ inX .
Consider the image f (x∞) ∈ Y .
Let V ∈ τY be such that f (x∞) ∈ V .
By continuity, U = f −1(V ) is open in X .
Observe that x∞ belongs to U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y , then f is continuous if and only if eachconvergent net xλ → x∞ satisfies f (xλ)→ f (x∞).
Proof: Suppose f : X → Y is continuous and let xλ → x∞ inX .
Consider the image f (x∞) ∈ Y .
Let V ∈ τY be such that f (x∞) ∈ V .
By continuity, U = f −1(V ) is open in X .
Observe that x∞ belongs to U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y , then f is continuous if and only if eachconvergent net xλ → x∞ satisfies f (xλ)→ f (x∞).
Proof: Suppose f : X → Y is continuous and let xλ → x∞ inX .
Consider the image f (x∞) ∈ Y .
Let V ∈ τY be such that f (x∞) ∈ V .
By continuity, U = f −1(V ) is open in X .
Observe that x∞ belongs to U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y , then f is continuous if and only if eachconvergent net xλ → x∞ satisfies f (xλ)→ f (x∞).
Proof: Suppose f : X → Y is continuous and let xλ → x∞ inX .
Consider the image f (x∞) ∈ Y .
Let V ∈ τY be such that f (x∞) ∈ V .
By continuity, U = f −1(V ) is open in X .
Observe that x∞ belongs to U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
There is a λU such that λU � λ implies xλ ∈ U.
It follows that λU � λ implies f (xλ) ∈ V .
Since V is arbitrary, this shows that f (xλ)→ f (x∞).
Conversely, suppose f is not continuous so that V ∈ τY hasA = f −1(V ) /∈ τX .
There is a net xλ ∈ X − A such that xλ → x∞ ∈ A.
The net f (xλ) is in Y − V , which is a closed set.
This net cannot converge to f (x∞) ∈ V .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
There is a λU such that λU � λ implies xλ ∈ U.
It follows that λU � λ implies f (xλ) ∈ V .
Since V is arbitrary, this shows that f (xλ)→ f (x∞).
Conversely, suppose f is not continuous so that V ∈ τY hasA = f −1(V ) /∈ τX .
There is a net xλ ∈ X − A such that xλ → x∞ ∈ A.
The net f (xλ) is in Y − V , which is a closed set.
This net cannot converge to f (x∞) ∈ V .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
There is a λU such that λU � λ implies xλ ∈ U.
It follows that λU � λ implies f (xλ) ∈ V .
Since V is arbitrary, this shows that f (xλ)→ f (x∞).
Conversely, suppose f is not continuous so that V ∈ τY hasA = f −1(V ) /∈ τX .
There is a net xλ ∈ X − A such that xλ → x∞ ∈ A.
The net f (xλ) is in Y − V , which is a closed set.
This net cannot converge to f (x∞) ∈ V .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
There is a λU such that λU � λ implies xλ ∈ U.
It follows that λU � λ implies f (xλ) ∈ V .
Since V is arbitrary, this shows that f (xλ)→ f (x∞).
Conversely, suppose f is not continuous so that V ∈ τY hasA = f −1(V ) /∈ τX .
There is a net xλ ∈ X − A such that xλ → x∞ ∈ A.
The net f (xλ) is in Y − V , which is a closed set.
This net cannot converge to f (x∞) ∈ V .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
There is a λU such that λU � λ implies xλ ∈ U.
It follows that λU � λ implies f (xλ) ∈ V .
Since V is arbitrary, this shows that f (xλ)→ f (x∞).
Conversely, suppose f is not continuous so that V ∈ τY hasA = f −1(V ) /∈ τX .
There is a net xλ ∈ X − A such that xλ → x∞ ∈ A.
The net f (xλ) is in Y − V , which is a closed set.
This net cannot converge to f (x∞) ∈ V .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
There is a λU such that λU � λ implies xλ ∈ U.
It follows that λU � λ implies f (xλ) ∈ V .
Since V is arbitrary, this shows that f (xλ)→ f (x∞).
Conversely, suppose f is not continuous so that V ∈ τY hasA = f −1(V ) /∈ τX .
There is a net xλ ∈ X − A such that xλ → x∞ ∈ A.
The net f (xλ) is in Y − V , which is a closed set.
This net cannot converge to f (x∞) ∈ V .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
There is a λU such that λU � λ implies xλ ∈ U.
It follows that λU � λ implies f (xλ) ∈ V .
Since V is arbitrary, this shows that f (xλ)→ f (x∞).
Conversely, suppose f is not continuous so that V ∈ τY hasA = f −1(V ) /∈ τX .
There is a net xλ ∈ X − A such that xλ → x∞ ∈ A.
The net f (xλ) is in Y − V , which is a closed set.
This net cannot converge to f (x∞) ∈ V .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A topological space is said to be sequentially compact if everysequence has a convergent subsequence.
A topological space is said to be compact if every open coverhas a finite subcover.
A metric space is compact if and only if it is sequentiallycompact.
There are compact topological spaces that are not sequentiallycompact.
There are sequentially compact topological spaces that are notcompact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A topological space is said to be sequentially compact if everysequence has a convergent subsequence.
A topological space is said to be compact if every open coverhas a finite subcover.
A metric space is compact if and only if it is sequentiallycompact.
There are compact topological spaces that are not sequentiallycompact.
There are sequentially compact topological spaces that are notcompact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A topological space is said to be sequentially compact if everysequence has a convergent subsequence.
A topological space is said to be compact if every open coverhas a finite subcover.
A metric space is compact if and only if it is sequentiallycompact.
There are compact topological spaces that are not sequentiallycompact.
There are sequentially compact topological spaces that are notcompact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A topological space is said to be sequentially compact if everysequence has a convergent subsequence.
A topological space is said to be compact if every open coverhas a finite subcover.
A metric space is compact if and only if it is sequentiallycompact.
There are compact topological spaces that are not sequentiallycompact.
There are sequentially compact topological spaces that are notcompact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A topological space is said to be sequentially compact if everysequence has a convergent subsequence.
A topological space is said to be compact if every open coverhas a finite subcover.
A metric space is compact if and only if it is sequentiallycompact.
There are compact topological spaces that are not sequentiallycompact.
There are sequentially compact topological spaces that are notcompact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose (D0,�0) and (D1,�1) are two directed sets.
A function f : D0 → D1 preserves order if x �0 y impliesf (x) �1 f (y), and is cofinal if to each y ∈ D1 there is anxy ∈ D0 such that y �1 f (xy ).
A net x : D1 → X has a subnet x ◦ f : D0 → X whenever fpreserves order and is cofinal.
A topological space X is compact if and only if each net has aconvergent subnet.
The following reasoning is flawed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose (D0,�0) and (D1,�1) are two directed sets.
A function f : D0 → D1 preserves order if x �0 y impliesf (x) �1 f (y), and is cofinal if to each y ∈ D1 there is anxy ∈ D0 such that y �1 f (xy ).
A net x : D1 → X has a subnet x ◦ f : D0 → X whenever fpreserves order and is cofinal.
A topological space X is compact if and only if each net has aconvergent subnet.
The following reasoning is flawed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose (D0,�0) and (D1,�1) are two directed sets.
A function f : D0 → D1 preserves order if x �0 y impliesf (x) �1 f (y), and is cofinal if to each y ∈ D1 there is anxy ∈ D0 such that y �1 f (xy ).
A net x : D1 → X has a subnet x ◦ f : D0 → X whenever fpreserves order and is cofinal.
A topological space X is compact if and only if each net has aconvergent subnet.
The following reasoning is flawed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose (D0,�0) and (D1,�1) are two directed sets.
A function f : D0 → D1 preserves order if x �0 y impliesf (x) �1 f (y), and is cofinal if to each y ∈ D1 there is anxy ∈ D0 such that y �1 f (xy ).
A net x : D1 → X has a subnet x ◦ f : D0 → X whenever fpreserves order and is cofinal.
A topological space X is compact if and only if each net has aconvergent subnet.
The following reasoning is flawed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose (D0,�0) and (D1,�1) are two directed sets.
A function f : D0 → D1 preserves order if x �0 y impliesf (x) �1 f (y), and is cofinal if to each y ∈ D1 there is anxy ∈ D0 such that y �1 f (xy ).
A net x : D1 → X has a subnet x ◦ f : D0 → X whenever fpreserves order and is cofinal.
A topological space X is compact if and only if each net has aconvergent subnet.
The following reasoning is flawed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
“A sequence may be regarded as a net.
Inside a compact space there is therefore a convergent subnet.
This produces a convergent subsequence, and hence everycompact space is sequentially compact.”
The subtlety is that a subnet of a sequence need not be asubsequence, f need not be 1-1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
“A sequence may be regarded as a net.
Inside a compact space there is therefore a convergent subnet.
This produces a convergent subsequence, and hence everycompact space is sequentially compact.”
The subtlety is that a subnet of a sequence need not be asubsequence, f need not be 1-1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
“A sequence may be regarded as a net.
Inside a compact space there is therefore a convergent subnet.
This produces a convergent subsequence, and hence everycompact space is sequentially compact.”
The subtlety is that a subnet of a sequence need not be asubsequence, f need not be 1-1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
“A sequence may be regarded as a net.
Inside a compact space there is therefore a convergent subnet.
This produces a convergent subsequence, and hence everycompact space is sequentially compact.”
The subtlety is that a subnet of a sequence need not be asubsequence, f need not be 1-1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is compact and let A ⊂ X be closed.
Consider a net x : D1 → A in X .
There is a convergent subnet x ◦ f : D0 → A.
Since A is closed, the limit must be in A.
The original net is arbitrary in A so it follows that A is compact.
Closed subsets of compact spaces are always compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is compact and let A ⊂ X be closed.
Consider a net x : D1 → A in X .
There is a convergent subnet x ◦ f : D0 → A.
Since A is closed, the limit must be in A.
The original net is arbitrary in A so it follows that A is compact.
Closed subsets of compact spaces are always compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is compact and let A ⊂ X be closed.
Consider a net x : D1 → A in X .
There is a convergent subnet x ◦ f : D0 → A.
Since A is closed, the limit must be in A.
The original net is arbitrary in A so it follows that A is compact.
Closed subsets of compact spaces are always compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is compact and let A ⊂ X be closed.
Consider a net x : D1 → A in X .
There is a convergent subnet x ◦ f : D0 → A.
Since A is closed, the limit must be in A.
The original net is arbitrary in A so it follows that A is compact.
Closed subsets of compact spaces are always compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is compact and let A ⊂ X be closed.
Consider a net x : D1 → A in X .
There is a convergent subnet x ◦ f : D0 → A.
Since A is closed, the limit must be in A.
The original net is arbitrary in A so it follows that A is compact.
Closed subsets of compact spaces are always compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose X is compact and let A ⊂ X be closed.
Consider a net x : D1 → A in X .
There is a convergent subnet x ◦ f : D0 → A.
Since A is closed, the limit must be in A.
The original net is arbitrary in A so it follows that A is compact.
Closed subsets of compact spaces are always compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Without nets the argument is as follows.
Let {Uα} be and open cover of A.
Write Uα = A ∩ Vα where Vα is open in X .
Now add the open set X − A to the collection of Vα.
This is an open cover of X .
Let Vα1 , . . . ,Vαn together with X − A be a finite subcover.
Toss out X − A and conclude that Uα1 , . . . ,Uαn is a finitesubcover of A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Without nets the argument is as follows.
Let {Uα} be and open cover of A.
Write Uα = A ∩ Vα where Vα is open in X .
Now add the open set X − A to the collection of Vα.
This is an open cover of X .
Let Vα1 , . . . ,Vαn together with X − A be a finite subcover.
Toss out X − A and conclude that Uα1 , . . . ,Uαn is a finitesubcover of A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Without nets the argument is as follows.
Let {Uα} be and open cover of A.
Write Uα = A ∩ Vα where Vα is open in X .
Now add the open set X − A to the collection of Vα.
This is an open cover of X .
Let Vα1 , . . . ,Vαn together with X − A be a finite subcover.
Toss out X − A and conclude that Uα1 , . . . ,Uαn is a finitesubcover of A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Without nets the argument is as follows.
Let {Uα} be and open cover of A.
Write Uα = A ∩ Vα where Vα is open in X .
Now add the open set X − A to the collection of Vα.
This is an open cover of X .
Let Vα1 , . . . ,Vαn together with X − A be a finite subcover.
Toss out X − A and conclude that Uα1 , . . . ,Uαn is a finitesubcover of A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Without nets the argument is as follows.
Let {Uα} be and open cover of A.
Write Uα = A ∩ Vα where Vα is open in X .
Now add the open set X − A to the collection of Vα.
This is an open cover of X .
Let Vα1 , . . . ,Vαn together with X − A be a finite subcover.
Toss out X − A and conclude that Uα1 , . . . ,Uαn is a finitesubcover of A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Without nets the argument is as follows.
Let {Uα} be and open cover of A.
Write Uα = A ∩ Vα where Vα is open in X .
Now add the open set X − A to the collection of Vα.
This is an open cover of X .
Let Vα1 , . . . ,Vαn together with X − A be a finite subcover.
Toss out X − A and conclude that Uα1 , . . . ,Uαn is a finitesubcover of A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Without nets the argument is as follows.
Let {Uα} be and open cover of A.
Write Uα = A ∩ Vα where Vα is open in X .
Now add the open set X − A to the collection of Vα.
This is an open cover of X .
Let Vα1 , . . . ,Vαn together with X − A be a finite subcover.
Toss out X − A and conclude that Uα1 , . . . ,Uαn is a finitesubcover of A.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that a net x : D → X converges to x∞ ∈ X if to eachopen set U ∈ τ such that x∞ ∈ U, there is a λU ∈ D such thatλU � λ implies xλ ∈ U.
This may be expressed by the phrase “the net x is eventually inU”.
A point x∞ ∈ X is a cluster point of the net x if to each openset U ∈ τ such that x∞ ∈ U it is true that given any λ0 ∈ Dthere is some λ ∈ D with λ0 � λ and xλ ∈ U.
This may be expressed by the phrase “the net x is frequently inU”.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that a net x : D → X converges to x∞ ∈ X if to eachopen set U ∈ τ such that x∞ ∈ U, there is a λU ∈ D such thatλU � λ implies xλ ∈ U.
This may be expressed by the phrase “the net x is eventually inU”.
A point x∞ ∈ X is a cluster point of the net x if to each openset U ∈ τ such that x∞ ∈ U it is true that given any λ0 ∈ Dthere is some λ ∈ D with λ0 � λ and xλ ∈ U.
This may be expressed by the phrase “the net x is frequently inU”.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that a net x : D → X converges to x∞ ∈ X if to eachopen set U ∈ τ such that x∞ ∈ U, there is a λU ∈ D such thatλU � λ implies xλ ∈ U.
This may be expressed by the phrase “the net x is eventually inU”.
A point x∞ ∈ X is a cluster point of the net x if to each openset U ∈ τ such that x∞ ∈ U it is true that given any λ0 ∈ Dthere is some λ ∈ D with λ0 � λ and xλ ∈ U.
This may be expressed by the phrase “the net x is frequently inU”.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that a net x : D → X converges to x∞ ∈ X if to eachopen set U ∈ τ such that x∞ ∈ U, there is a λU ∈ D such thatλU � λ implies xλ ∈ U.
This may be expressed by the phrase “the net x is eventually inU”.
A point x∞ ∈ X is a cluster point of the net x if to each openset U ∈ τ such that x∞ ∈ U it is true that given any λ0 ∈ Dthere is some λ ∈ D with λ0 � λ and xλ ∈ U.
This may be expressed by the phrase “the net x is frequently inU”.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A collection of sets with the property that any finitesub-collection has nonempty intersection is said to satisfy thefinite intersection property. Theorem: The following areequivalent properties of a topological space (X , τ).
(1) Each net has a convergent subnet.
(2) Each net has a cluster point.
(3) Each open cover has, a finite subcover.
(4) Each collection of closed sets that satisfies the finiteintersection property has nonempty intersection.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A collection of sets with the property that any finitesub-collection has nonempty intersection is said to satisfy thefinite intersection property. Theorem: The following areequivalent properties of a topological space (X , τ).
(1) Each net has a convergent subnet.
(2) Each net has a cluster point.
(3) Each open cover has, a finite subcover.
(4) Each collection of closed sets that satisfies the finiteintersection property has nonempty intersection.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A collection of sets with the property that any finitesub-collection has nonempty intersection is said to satisfy thefinite intersection property. Theorem: The following areequivalent properties of a topological space (X , τ).
(1) Each net has a convergent subnet.
(2) Each net has a cluster point.
(3) Each open cover has, a finite subcover.
(4) Each collection of closed sets that satisfies the finiteintersection property has nonempty intersection.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A collection of sets with the property that any finitesub-collection has nonempty intersection is said to satisfy thefinite intersection property. Theorem: The following areequivalent properties of a topological space (X , τ).
(1) Each net has a convergent subnet.
(2) Each net has a cluster point.
(3) Each open cover has, a finite subcover.
(4) Each collection of closed sets that satisfies the finiteintersection property has nonempty intersection.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A collection of sets with the property that any finitesub-collection has nonempty intersection is said to satisfy thefinite intersection property. Theorem: The following areequivalent properties of a topological space (X , τ).
(1) Each net has a convergent subnet.
(2) Each net has a cluster point.
(3) Each open cover has, a finite subcover.
(4) Each collection of closed sets that satisfies the finiteintersection property has nonempty intersection.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Proof:
(1)⇒(2)
Suppose there is a subnet of x : D1 → X converging tox∞ ∈ X .
Let U be an open set containing x∞.
Let y : D0 → X be the subnet via f : D0 → D1.
There is a λ0U ∈ D0 such that λ0
U � λ0 ⇒ yλ0 ∈ U.
Given any λ1 ∈ D1, there is a λ0λ1 ∈ D0 such that
λ0λ1 � λ0 ⇒ λ1 � f (λ0).
There is now a λ00 ∈ D0 such that λ0
U � λ00 and λ0
λ1 � λ00.
Let λ10 = f (λ0
0) and note that xλ10∈ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(1)
Suppose x∞ is a cluster point of x : D → X .
Define the set of pairs
R = {(λ,U) | x∞ ∈ U ∈ τ, xλ ∈ U} .
Write (λ0,U0) �R (λ1,U1) if and only if λ0 � λ1 and U0 ⊃ U1.
Now R is a directed set.
Let f : R → D be given by f (λ,U) = λ.
Let y = x ◦ f , and consider any open set U containing x∞.
Choose any λ0 ∈ D such that xλ0 ∈ U.
If (λ0,U) �R (λ,U0), then
y(λ,U0) = xλ ∈ U0 ⊂ U.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(3)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets such that each finitesubcollection has nonempty intersection.
If ⋂γ∈Γ
Cγ = ∅,
then ⋃γ∈Γ
C cγ = X .
It follows thatC cγ1∪ · · · ∪ C c
γn = X ,
and henceCγ1 ∩ · · · ∩ Cγn = ∅.
This is a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(3)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets such that each finitesubcollection has nonempty intersection.
If ⋂γ∈Γ
Cγ = ∅,
then ⋃γ∈Γ
C cγ = X .
It follows thatC cγ1∪ · · · ∪ C c
γn = X ,
and henceCγ1 ∩ · · · ∩ Cγn = ∅.
This is a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(3)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets such that each finitesubcollection has nonempty intersection.
If ⋂γ∈Γ
Cγ = ∅,
then ⋃γ∈Γ
C cγ = X .
It follows thatC cγ1∪ · · · ∪ C c
γn = X ,
and henceCγ1 ∩ · · · ∩ Cγn = ∅.
This is a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(3)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets such that each finitesubcollection has nonempty intersection.
If ⋂γ∈Γ
Cγ = ∅,
then ⋃γ∈Γ
C cγ = X .
It follows thatC cγ1∪ · · · ∪ C c
γn = X ,
and henceCγ1 ∩ · · · ∩ Cγn = ∅.
This is a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(3)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets such that each finitesubcollection has nonempty intersection.
If ⋂γ∈Γ
Cγ = ∅,
then ⋃γ∈Γ
C cγ = X .
It follows thatC cγ1∪ · · · ∪ C c
γn = X ,
and henceCγ1 ∩ · · · ∩ Cγn = ∅.
This is a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(3)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets such that each finitesubcollection has nonempty intersection.
If ⋂γ∈Γ
Cγ = ∅,
then ⋃γ∈Γ
C cγ = X .
It follows thatC cγ1∪ · · · ∪ C c
γn = X ,
and henceCγ1 ∩ · · · ∩ Cγn = ∅.
This is a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(3)
Let {Uγ}γ∈Γ be a collection of open sets such that⋃γ∈Γ
Uγ = X .
Suppose there is no finite subcover, then
Uγ1 ∪ · · · ∪ Uγn 6= X ,
and henceUcγ1∩ · · · ∩ Uc
γn 6= ∅.
It follows that ⋂γ∈Γ
Ucγ 6= ∅.
Taking complements yields a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(3)
Let {Uγ}γ∈Γ be a collection of open sets such that⋃γ∈Γ
Uγ = X .
Suppose there is no finite subcover, then
Uγ1 ∪ · · · ∪ Uγn 6= X ,
and henceUcγ1∩ · · · ∩ Uc
γn 6= ∅.
It follows that ⋂γ∈Γ
Ucγ 6= ∅.
Taking complements yields a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(3)
Let {Uγ}γ∈Γ be a collection of open sets such that⋃γ∈Γ
Uγ = X .
Suppose there is no finite subcover, then
Uγ1 ∪ · · · ∪ Uγn 6= X ,
and henceUcγ1∩ · · · ∩ Uc
γn 6= ∅.
It follows that ⋂γ∈Γ
Ucγ 6= ∅.
Taking complements yields a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(3)
Let {Uγ}γ∈Γ be a collection of open sets such that⋃γ∈Γ
Uγ = X .
Suppose there is no finite subcover, then
Uγ1 ∪ · · · ∪ Uγn 6= X ,
and henceUcγ1∩ · · · ∩ Uc
γn 6= ∅.
It follows that ⋂γ∈Γ
Ucγ 6= ∅.
Taking complements yields a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(3)
Let {Uγ}γ∈Γ be a collection of open sets such that⋃γ∈Γ
Uγ = X .
Suppose there is no finite subcover, then
Uγ1 ∪ · · · ∪ Uγn 6= X ,
and henceUcγ1∩ · · · ∩ Uc
γn 6= ∅.
It follows that ⋂γ∈Γ
Ucγ 6= ∅.
Taking complements yields a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(3)
Let {Uγ}γ∈Γ be a collection of open sets such that⋃γ∈Γ
Uγ = X .
Suppose there is no finite subcover, then
Uγ1 ∪ · · · ∪ Uγn 6= X ,
and henceUcγ1∩ · · · ∩ Uc
γn 6= ∅.
It follows that ⋂γ∈Γ
Ucγ 6= ∅.
Taking complements yields a contradiction.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(2)⇒(4)
Let {Cγ}γ∈Γ be a collection of closed sets that satisfies thefinite intersection property.
Let
D =
C
∣∣∣∣∣∣ C =⋂γ∈F
Cγ , F ⊂ Γ, F finite
.
Write C0 � C1 if C0 ⊃ C1.
Define x : D → X by choosing xC ∈ C . Let x∞ ∈ X be acluster point of x .
If C � C0, then xC ∈ C0 ⊂ C .
It follows that x∞ ∈ C , because otherwise x is eventually in theopen set C c .
It must be thatx∞ ∈
⋂γ∈Γ
Cγ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(2)
Let x : D → X be a net.
Define the sets
Aλ = {xλ0 ∈ X | λ � λ0} .
The collection {A−λ}λ∈D
satisfies the finite intersection property.
Letx∞ ∈
⋂λ∈D
A−λ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(2)
Let x : D → X be a net.
Define the sets
Aλ = {xλ0 ∈ X | λ � λ0} .
The collection {A−λ}λ∈D
satisfies the finite intersection property.
Letx∞ ∈
⋂λ∈D
A−λ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(2)
Let x : D → X be a net.
Define the sets
Aλ = {xλ0 ∈ X | λ � λ0} .
The collection {A−λ}λ∈D
satisfies the finite intersection property.
Letx∞ ∈
⋂λ∈D
A−λ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(2)
Let x : D → X be a net.
Define the sets
Aλ = {xλ0 ∈ X | λ � λ0} .
The collection {A−λ}λ∈D
satisfies the finite intersection property.
Letx∞ ∈
⋂λ∈D
A−λ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(2)
Let x : D → X be a net.
Define the sets
Aλ = {xλ0 ∈ X | λ � λ0} .
The collection {A−λ}λ∈D
satisfies the finite intersection property.
Letx∞ ∈
⋂λ∈D
A−λ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
(4)⇒(2)
Let x : D → X be a net.
Define the sets
Aλ = {xλ0 ∈ X | λ � λ0} .
The collection {A−λ}λ∈D
satisfies the finite intersection property.
Letx∞ ∈
⋂λ∈D
A−λ .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
If x∞ is not a cluster point of x , then there is some open set Ucontaining x∞ and some λ ∈ D such that
λ � λ0 ⇒ xλ0 /∈ U.
It follows thatAλ ∩ U = ∅,
and hencex∞ /∈ A−λ
which is impossible.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
If x∞ is not a cluster point of x , then there is some open set Ucontaining x∞ and some λ ∈ D such that
λ � λ0 ⇒ xλ0 /∈ U.
It follows thatAλ ∩ U = ∅,
and hencex∞ /∈ A−λ
which is impossible.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Compactness in Rn is the same as closed and bounded.
Compactness in a metric space is the same as complete andtotally bounded.
Recall that totally bounded means for each r > 0 it should bepossible to cover the whole space by a finite number of openballs of radius r .
Suppose X is a topological space and f : X → R.
Let α = infX f (x).
It is possible to have α = −∞.
Assume α > −∞.
It is possible that there is no x ∈ X such that f (x) = α.
There are minimizing sequences xk with f (xk) < α + 1k .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A sequence is an example of a net.
If X is compact, then there is a convergent subnet.
The trouble is that a subnet need not be a subsequence.
There is a directed set D and a map ϕ : D → N.
There is some x∞ ∈ X such that the subnet converges to x∞.
Now assume that f (x∞) > α.
This certainly seems possible unless there is some additionalassumption on f .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A sequence is an example of a net.
If X is compact, then there is a convergent subnet.
The trouble is that a subnet need not be a subsequence.
There is a directed set D and a map ϕ : D → N.
There is some x∞ ∈ X such that the subnet converges to x∞.
Now assume that f (x∞) > α.
This certainly seems possible unless there is some additionalassumption on f .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A sequence is an example of a net.
If X is compact, then there is a convergent subnet.
The trouble is that a subnet need not be a subsequence.
There is a directed set D and a map ϕ : D → N.
There is some x∞ ∈ X such that the subnet converges to x∞.
Now assume that f (x∞) > α.
This certainly seems possible unless there is some additionalassumption on f .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A sequence is an example of a net.
If X is compact, then there is a convergent subnet.
The trouble is that a subnet need not be a subsequence.
There is a directed set D and a map ϕ : D → N.
There is some x∞ ∈ X such that the subnet converges to x∞.
Now assume that f (x∞) > α.
This certainly seems possible unless there is some additionalassumption on f .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A sequence is an example of a net.
If X is compact, then there is a convergent subnet.
The trouble is that a subnet need not be a subsequence.
There is a directed set D and a map ϕ : D → N.
There is some x∞ ∈ X such that the subnet converges to x∞.
Now assume that f (x∞) > α.
This certainly seems possible unless there is some additionalassumption on f .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A sequence is an example of a net.
If X is compact, then there is a convergent subnet.
The trouble is that a subnet need not be a subsequence.
There is a directed set D and a map ϕ : D → N.
There is some x∞ ∈ X such that the subnet converges to x∞.
Now assume that f (x∞) > α.
This certainly seems possible unless there is some additionalassumption on f .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A sequence is an example of a net.
If X is compact, then there is a convergent subnet.
The trouble is that a subnet need not be a subsequence.
There is a directed set D and a map ϕ : D → N.
There is some x∞ ∈ X such that the subnet converges to x∞.
Now assume that f (x∞) > α.
This certainly seems possible unless there is some additionalassumption on f .
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A possible choice is to assume Uβ = {x : f (x) > β} is open nomatter what the value of β ∈ R is.
This condition is known as lower semi-continuity.
Choose β = 12 (f (x∞) + α) and observe that x∞ ∈ Uβ.
The subnet must be in Uβ eventually, which creates acontradiction when f (x∞) > α.
The only resolution is to have f (x∞) = α, which may expressedby asserting that the minimum is attained at x∞ ∈ X .
The combination compact space and semi-continuous functionis therefore an attractive way to guarantee the existence of anoptimal point.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A possible choice is to assume Uβ = {x : f (x) > β} is open nomatter what the value of β ∈ R is.
This condition is known as lower semi-continuity.
Choose β = 12 (f (x∞) + α) and observe that x∞ ∈ Uβ.
The subnet must be in Uβ eventually, which creates acontradiction when f (x∞) > α.
The only resolution is to have f (x∞) = α, which may expressedby asserting that the minimum is attained at x∞ ∈ X .
The combination compact space and semi-continuous functionis therefore an attractive way to guarantee the existence of anoptimal point.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A possible choice is to assume Uβ = {x : f (x) > β} is open nomatter what the value of β ∈ R is.
This condition is known as lower semi-continuity.
Choose β = 12 (f (x∞) + α) and observe that x∞ ∈ Uβ.
The subnet must be in Uβ eventually, which creates acontradiction when f (x∞) > α.
The only resolution is to have f (x∞) = α, which may expressedby asserting that the minimum is attained at x∞ ∈ X .
The combination compact space and semi-continuous functionis therefore an attractive way to guarantee the existence of anoptimal point.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A possible choice is to assume Uβ = {x : f (x) > β} is open nomatter what the value of β ∈ R is.
This condition is known as lower semi-continuity.
Choose β = 12 (f (x∞) + α) and observe that x∞ ∈ Uβ.
The subnet must be in Uβ eventually, which creates acontradiction when f (x∞) > α.
The only resolution is to have f (x∞) = α, which may expressedby asserting that the minimum is attained at x∞ ∈ X .
The combination compact space and semi-continuous functionis therefore an attractive way to guarantee the existence of anoptimal point.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A possible choice is to assume Uβ = {x : f (x) > β} is open nomatter what the value of β ∈ R is.
This condition is known as lower semi-continuity.
Choose β = 12 (f (x∞) + α) and observe that x∞ ∈ Uβ.
The subnet must be in Uβ eventually, which creates acontradiction when f (x∞) > α.
The only resolution is to have f (x∞) = α, which may expressedby asserting that the minimum is attained at x∞ ∈ X .
The combination compact space and semi-continuous functionis therefore an attractive way to guarantee the existence of anoptimal point.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A possible choice is to assume Uβ = {x : f (x) > β} is open nomatter what the value of β ∈ R is.
This condition is known as lower semi-continuity.
Choose β = 12 (f (x∞) + α) and observe that x∞ ∈ Uβ.
The subnet must be in Uβ eventually, which creates acontradiction when f (x∞) > α.
The only resolution is to have f (x∞) = α, which may expressedby asserting that the minimum is attained at x∞ ∈ X .
The combination compact space and semi-continuous functionis therefore an attractive way to guarantee the existence of anoptimal point.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This motivates the idea of altering the topology to make iteasier to be compact.
The trouble is that this suggests ‘reducing the number’ of opensets to reduce the number of open cover one must consider, butthis would make it more difficult to be lower semi-continuous.
The space [0, 1) ⊂ R is not compact since the open cover[0, 1− 1
k+2 ) with k ∈ N has no finite subcover.
Let φ : [0, 1]→ [0, 1) be given by φ(1) = 0 and then use yourfavorite 1− 1 and onto function from [0, 1) to (0, 1).
Since [0, 1] is compact, a topological property, this implies thateach such function is not continuous or its inverse is notcontinuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This motivates the idea of altering the topology to make iteasier to be compact.
The trouble is that this suggests ‘reducing the number’ of opensets to reduce the number of open cover one must consider, butthis would make it more difficult to be lower semi-continuous.
The space [0, 1) ⊂ R is not compact since the open cover[0, 1− 1
k+2 ) with k ∈ N has no finite subcover.
Let φ : [0, 1]→ [0, 1) be given by φ(1) = 0 and then use yourfavorite 1− 1 and onto function from [0, 1) to (0, 1).
Since [0, 1] is compact, a topological property, this implies thateach such function is not continuous or its inverse is notcontinuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This motivates the idea of altering the topology to make iteasier to be compact.
The trouble is that this suggests ‘reducing the number’ of opensets to reduce the number of open cover one must consider, butthis would make it more difficult to be lower semi-continuous.
The space [0, 1) ⊂ R is not compact since the open cover[0, 1− 1
k+2 ) with k ∈ N has no finite subcover.
Let φ : [0, 1]→ [0, 1) be given by φ(1) = 0 and then use yourfavorite 1− 1 and onto function from [0, 1) to (0, 1).
Since [0, 1] is compact, a topological property, this implies thateach such function is not continuous or its inverse is notcontinuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This motivates the idea of altering the topology to make iteasier to be compact.
The trouble is that this suggests ‘reducing the number’ of opensets to reduce the number of open cover one must consider, butthis would make it more difficult to be lower semi-continuous.
The space [0, 1) ⊂ R is not compact since the open cover[0, 1− 1
k+2 ) with k ∈ N has no finite subcover.
Let φ : [0, 1]→ [0, 1) be given by φ(1) = 0 and then use yourfavorite 1− 1 and onto function from [0, 1) to (0, 1).
Since [0, 1] is compact, a topological property, this implies thateach such function is not continuous or its inverse is notcontinuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This motivates the idea of altering the topology to make iteasier to be compact.
The trouble is that this suggests ‘reducing the number’ of opensets to reduce the number of open cover one must consider, butthis would make it more difficult to be lower semi-continuous.
The space [0, 1) ⊂ R is not compact since the open cover[0, 1− 1
k+2 ) with k ∈ N has no finite subcover.
Let φ : [0, 1]→ [0, 1) be given by φ(1) = 0 and then use yourfavorite 1− 1 and onto function from [0, 1) to (0, 1).
Since [0, 1] is compact, a topological property, this implies thateach such function is not continuous or its inverse is notcontinuous.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This illustrates a common way to demonstrate that twotopological spaces are not homeomorphic: exhibit a topologicalproperty that is not shared by both spaces.
The unit circle S1 = {(x , y) ∈ R2 : x2 + y2 = 1} is compactwhereas the real line R is not compact, so as expected the twospaces are not homeomorphic.
Since sin−1([−1, 1]) = R, the inverse image of a compact setunder a continuous map need not be compact.
It is however true that the continuous image of a compact setmust be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This illustrates a common way to demonstrate that twotopological spaces are not homeomorphic: exhibit a topologicalproperty that is not shared by both spaces.
The unit circle S1 = {(x , y) ∈ R2 : x2 + y2 = 1} is compactwhereas the real line R is not compact, so as expected the twospaces are not homeomorphic.
Since sin−1([−1, 1]) = R, the inverse image of a compact setunder a continuous map need not be compact.
It is however true that the continuous image of a compact setmust be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This illustrates a common way to demonstrate that twotopological spaces are not homeomorphic: exhibit a topologicalproperty that is not shared by both spaces.
The unit circle S1 = {(x , y) ∈ R2 : x2 + y2 = 1} is compactwhereas the real line R is not compact, so as expected the twospaces are not homeomorphic.
Since sin−1([−1, 1]) = R, the inverse image of a compact setunder a continuous map need not be compact.
It is however true that the continuous image of a compact setmust be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
This illustrates a common way to demonstrate that twotopological spaces are not homeomorphic: exhibit a topologicalproperty that is not shared by both spaces.
The unit circle S1 = {(x , y) ∈ R2 : x2 + y2 = 1} is compactwhereas the real line R is not compact, so as expected the twospaces are not homeomorphic.
Since sin−1([−1, 1]) = R, the inverse image of a compact setunder a continuous map need not be compact.
It is however true that the continuous image of a compact setmust be compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In terms of nets the reasoning is a follows.
Assume f : X → Y is continuous and A ⊂ X compact.
Let yλ be a net in f (A).
Consider any associated net xλ ∈ A such that f (xλ) = yλ.
There is a convergent subnet of xλ that converges to x∞ ∈ A,and its f image is a subnet of yλ.
The continuity of f guarantees that the image subnetconverges to f (x∞) ∈ f (A).
It follows that f (A) is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In terms of nets the reasoning is a follows.
Assume f : X → Y is continuous and A ⊂ X compact.
Let yλ be a net in f (A).
Consider any associated net xλ ∈ A such that f (xλ) = yλ.
There is a convergent subnet of xλ that converges to x∞ ∈ A,and its f image is a subnet of yλ.
The continuity of f guarantees that the image subnetconverges to f (x∞) ∈ f (A).
It follows that f (A) is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In terms of nets the reasoning is a follows.
Assume f : X → Y is continuous and A ⊂ X compact.
Let yλ be a net in f (A).
Consider any associated net xλ ∈ A such that f (xλ) = yλ.
There is a convergent subnet of xλ that converges to x∞ ∈ A,and its f image is a subnet of yλ.
The continuity of f guarantees that the image subnetconverges to f (x∞) ∈ f (A).
It follows that f (A) is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In terms of nets the reasoning is a follows.
Assume f : X → Y is continuous and A ⊂ X compact.
Let yλ be a net in f (A).
Consider any associated net xλ ∈ A such that f (xλ) = yλ.
There is a convergent subnet of xλ that converges to x∞ ∈ A,and its f image is a subnet of yλ.
The continuity of f guarantees that the image subnetconverges to f (x∞) ∈ f (A).
It follows that f (A) is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In terms of nets the reasoning is a follows.
Assume f : X → Y is continuous and A ⊂ X compact.
Let yλ be a net in f (A).
Consider any associated net xλ ∈ A such that f (xλ) = yλ.
There is a convergent subnet of xλ that converges to x∞ ∈ A,and its f image is a subnet of yλ.
The continuity of f guarantees that the image subnetconverges to f (x∞) ∈ f (A).
It follows that f (A) is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In terms of nets the reasoning is a follows.
Assume f : X → Y is continuous and A ⊂ X compact.
Let yλ be a net in f (A).
Consider any associated net xλ ∈ A such that f (xλ) = yλ.
There is a convergent subnet of xλ that converges to x∞ ∈ A,and its f image is a subnet of yλ.
The continuity of f guarantees that the image subnetconverges to f (x∞) ∈ f (A).
It follows that f (A) is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
In terms of nets the reasoning is a follows.
Assume f : X → Y is continuous and A ⊂ X compact.
Let yλ be a net in f (A).
Consider any associated net xλ ∈ A such that f (xλ) = yλ.
There is a convergent subnet of xλ that converges to x∞ ∈ A,and its f image is a subnet of yλ.
The continuity of f guarantees that the image subnetconverges to f (x∞) ∈ f (A).
It follows that f (A) is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and C ⊂ Y closed.
Let xλ → x∞ with xλ ∈ f −1(C ).
The continuity implies f (xλ)→ f (x∞).
Since f (xλ) ∈ C and C is closed, it follows that f (x∞) ∈ C .
The convergent net xλ ∈ f −1(C ) is arbitrary and x∞ ∈ f −1(C ),so f −1(C ) is closed.
For continuous functions the inverse image of a closed set isalways closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and C ⊂ Y closed.
Let xλ → x∞ with xλ ∈ f −1(C ).
The continuity implies f (xλ)→ f (x∞).
Since f (xλ) ∈ C and C is closed, it follows that f (x∞) ∈ C .
The convergent net xλ ∈ f −1(C ) is arbitrary and x∞ ∈ f −1(C ),so f −1(C ) is closed.
For continuous functions the inverse image of a closed set isalways closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and C ⊂ Y closed.
Let xλ → x∞ with xλ ∈ f −1(C ).
The continuity implies f (xλ)→ f (x∞).
Since f (xλ) ∈ C and C is closed, it follows that f (x∞) ∈ C .
The convergent net xλ ∈ f −1(C ) is arbitrary and x∞ ∈ f −1(C ),so f −1(C ) is closed.
For continuous functions the inverse image of a closed set isalways closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and C ⊂ Y closed.
Let xλ → x∞ with xλ ∈ f −1(C ).
The continuity implies f (xλ)→ f (x∞).
Since f (xλ) ∈ C and C is closed, it follows that f (x∞) ∈ C .
The convergent net xλ ∈ f −1(C ) is arbitrary and x∞ ∈ f −1(C ),so f −1(C ) is closed.
For continuous functions the inverse image of a closed set isalways closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and C ⊂ Y closed.
Let xλ → x∞ with xλ ∈ f −1(C ).
The continuity implies f (xλ)→ f (x∞).
Since f (xλ) ∈ C and C is closed, it follows that f (x∞) ∈ C .
The convergent net xλ ∈ f −1(C ) is arbitrary and x∞ ∈ f −1(C ),so f −1(C ) is closed.
For continuous functions the inverse image of a closed set isalways closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → Y is continuous and C ⊂ Y closed.
Let xλ → x∞ with xλ ∈ f −1(C ).
The continuity implies f (xλ)→ f (x∞).
Since f (xλ) ∈ C and C is closed, it follows that f (x∞) ∈ C .
The convergent net xλ ∈ f −1(C ) is arbitrary and x∞ ∈ f −1(C ),so f −1(C ) is closed.
For continuous functions the inverse image of a closed set isalways closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose f : X → X is continuous.
Let F = {x ∈ X | f (x) = x}.
This is the collection of fixed points of f .
Suppose xλ ∈ F is a convergent net that satisfies xλ → x∞.
By continuity it follows that f (xλ)→ f (x∞), and hencexλ → f (x∞).
If it is assumed that X is Hausdorff, then limits are unique andf (x∞) = x∞.
It follows that x∞ is a fixed point so that x∞ ∈ F , and hence Fis closed.
In Hausdorff spaces the set of fixed points of a continuousfunction is always closed.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let X be a set and T the collection of all topologies on X .
Consider a subset S of a given topology τ , so S is a collectionof open sets with S ⊂ τ rather than an open set, which in thiscase would mean S ∈ τ .
The discrete topology contains S .
LetτS =
⋂τ∈T &S⊂τ
τ.
It turns out that τS is a topology on X .
The collection S is said to be a subbasis for the topology τS .
The topology τS consists of arbitrary unions of finiteintersections of members of S and the whole set if not includedalready.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let X be a set and T the collection of all topologies on X .
Consider a subset S of a given topology τ , so S is a collectionof open sets with S ⊂ τ rather than an open set, which in thiscase would mean S ∈ τ .
The discrete topology contains S .
LetτS =
⋂τ∈T &S⊂τ
τ.
It turns out that τS is a topology on X .
The collection S is said to be a subbasis for the topology τS .
The topology τS consists of arbitrary unions of finiteintersections of members of S and the whole set if not includedalready.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let X be a set and T the collection of all topologies on X .
Consider a subset S of a given topology τ , so S is a collectionof open sets with S ⊂ τ rather than an open set, which in thiscase would mean S ∈ τ .
The discrete topology contains S .
LetτS =
⋂τ∈T &S⊂τ
τ.
It turns out that τS is a topology on X .
The collection S is said to be a subbasis for the topology τS .
The topology τS consists of arbitrary unions of finiteintersections of members of S and the whole set if not includedalready.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let X be a set and T the collection of all topologies on X .
Consider a subset S of a given topology τ , so S is a collectionof open sets with S ⊂ τ rather than an open set, which in thiscase would mean S ∈ τ .
The discrete topology contains S .
LetτS =
⋂τ∈T &S⊂τ
τ.
It turns out that τS is a topology on X .
The collection S is said to be a subbasis for the topology τS .
The topology τS consists of arbitrary unions of finiteintersections of members of S and the whole set if not includedalready.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let X be a set and T the collection of all topologies on X .
Consider a subset S of a given topology τ , so S is a collectionof open sets with S ⊂ τ rather than an open set, which in thiscase would mean S ∈ τ .
The discrete topology contains S .
LetτS =
⋂τ∈T &S⊂τ
τ.
It turns out that τS is a topology on X .
The collection S is said to be a subbasis for the topology τS .
The topology τS consists of arbitrary unions of finiteintersections of members of S and the whole set if not includedalready.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let X be a set and T the collection of all topologies on X .
Consider a subset S of a given topology τ , so S is a collectionof open sets with S ⊂ τ rather than an open set, which in thiscase would mean S ∈ τ .
The discrete topology contains S .
LetτS =
⋂τ∈T &S⊂τ
τ.
It turns out that τS is a topology on X .
The collection S is said to be a subbasis for the topology τS .
The topology τS consists of arbitrary unions of finiteintersections of members of S and the whole set if not includedalready.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let X be a set and T the collection of all topologies on X .
Consider a subset S of a given topology τ , so S is a collectionof open sets with S ⊂ τ rather than an open set, which in thiscase would mean S ∈ τ .
The discrete topology contains S .
LetτS =
⋂τ∈T &S⊂τ
τ.
It turns out that τS is a topology on X .
The collection S is said to be a subbasis for the topology τS .
The topology τS consists of arbitrary unions of finiteintersections of members of S and the whole set if not includedalready.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Given (X , τ), a collection B ⊂ τ is said to be a basis for τ ifeach open set is the union of members in B.
The collection of open intervals in R is a basis for the standardtopology of R.
The open rays in R is not a basis although it is a subbasis forthe standard topology of R.
Some lines of reasoning may simplify as the conditions involvedare only necessary to verify on a basis or maybe even a subbasis.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Given (X , τ), a collection B ⊂ τ is said to be a basis for τ ifeach open set is the union of members in B.
The collection of open intervals in R is a basis for the standardtopology of R.
The open rays in R is not a basis although it is a subbasis forthe standard topology of R.
Some lines of reasoning may simplify as the conditions involvedare only necessary to verify on a basis or maybe even a subbasis.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Given (X , τ), a collection B ⊂ τ is said to be a basis for τ ifeach open set is the union of members in B.
The collection of open intervals in R is a basis for the standardtopology of R.
The open rays in R is not a basis although it is a subbasis forthe standard topology of R.
Some lines of reasoning may simplify as the conditions involvedare only necessary to verify on a basis or maybe even a subbasis.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Given (X , τ), a collection B ⊂ τ is said to be a basis for τ ifeach open set is the union of members in B.
The collection of open intervals in R is a basis for the standardtopology of R.
The open rays in R is not a basis although it is a subbasis forthe standard topology of R.
Some lines of reasoning may simplify as the conditions involvedare only necessary to verify on a basis or maybe even a subbasis.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let (X , τ0) and (X , τ1) be two topological spaces with thesame underlying set of points X .
If τ0 ⊂ τ1, then τ0 is said to be weaker than τ1, and τ1 is saidto be stronger than τ0.
A pair of topologies need not be comparable in this sense.
The term finer is sometimes used in place of stronger, andcoarser is sometimes used in place of weaker.
A collection of functions F = {fα : S → (Xα, τα)} may be usedto introduce a topology on the set S , the domain of thefunctions.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let (X , τ0) and (X , τ1) be two topological spaces with thesame underlying set of points X .
If τ0 ⊂ τ1, then τ0 is said to be weaker than τ1, and τ1 is saidto be stronger than τ0.
A pair of topologies need not be comparable in this sense.
The term finer is sometimes used in place of stronger, andcoarser is sometimes used in place of weaker.
A collection of functions F = {fα : S → (Xα, τα)} may be usedto introduce a topology on the set S , the domain of thefunctions.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let (X , τ0) and (X , τ1) be two topological spaces with thesame underlying set of points X .
If τ0 ⊂ τ1, then τ0 is said to be weaker than τ1, and τ1 is saidto be stronger than τ0.
A pair of topologies need not be comparable in this sense.
The term finer is sometimes used in place of stronger, andcoarser is sometimes used in place of weaker.
A collection of functions F = {fα : S → (Xα, τα)} may be usedto introduce a topology on the set S , the domain of thefunctions.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let (X , τ0) and (X , τ1) be two topological spaces with thesame underlying set of points X .
If τ0 ⊂ τ1, then τ0 is said to be weaker than τ1, and τ1 is saidto be stronger than τ0.
A pair of topologies need not be comparable in this sense.
The term finer is sometimes used in place of stronger, andcoarser is sometimes used in place of weaker.
A collection of functions F = {fα : S → (Xα, τα)} may be usedto introduce a topology on the set S , the domain of thefunctions.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let (X , τ0) and (X , τ1) be two topological spaces with thesame underlying set of points X .
If τ0 ⊂ τ1, then τ0 is said to be weaker than τ1, and τ1 is saidto be stronger than τ0.
A pair of topologies need not be comparable in this sense.
The term finer is sometimes used in place of stronger, andcoarser is sometimes used in place of weaker.
A collection of functions F = {fα : S → (Xα, τα)} may be usedto introduce a topology on the set S , the domain of thefunctions.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Observe that the set of subsets, P(S), would render each fαcontinuous.
Let τF be the intersection of all topologies that render each fαcontinuous.
The topology τF is known as the weak topology on S inducedby F .
One has sλ → s∞ if and only if fα(sλ)→ fα(s∞) for all α.
It may be that S already has a topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Observe that the set of subsets, P(S), would render each fαcontinuous.
Let τF be the intersection of all topologies that render each fαcontinuous.
The topology τF is known as the weak topology on S inducedby F .
One has sλ → s∞ if and only if fα(sλ)→ fα(s∞) for all α.
It may be that S already has a topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Observe that the set of subsets, P(S), would render each fαcontinuous.
Let τF be the intersection of all topologies that render each fαcontinuous.
The topology τF is known as the weak topology on S inducedby F .
One has sλ → s∞ if and only if fα(sλ)→ fα(s∞) for all α.
It may be that S already has a topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Observe that the set of subsets, P(S), would render each fαcontinuous.
Let τF be the intersection of all topologies that render each fαcontinuous.
The topology τF is known as the weak topology on S inducedby F .
One has sλ → s∞ if and only if fα(sλ)→ fα(s∞) for all α.
It may be that S already has a topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Observe that the set of subsets, P(S), would render each fαcontinuous.
Let τF be the intersection of all topologies that render each fαcontinuous.
The topology τF is known as the weak topology on S inducedby F .
One has sλ → s∞ if and only if fα(sλ)→ fα(s∞) for all α.
It may be that S already has a topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let S be a normed vector space, and let F be the collection ofcontinuous linear functionals on S .
There are now two topologies on S , the metric topologyinduced by the norm and the weak topology induced by F .
The terminology is consistent here since it is possible to showthat the weak topology in fact is weaker than then normtopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let S be a normed vector space, and let F be the collection ofcontinuous linear functionals on S .
There are now two topologies on S , the metric topologyinduced by the norm and the weak topology induced by F .
The terminology is consistent here since it is possible to showthat the weak topology in fact is weaker than then normtopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let S be a normed vector space, and let F be the collection ofcontinuous linear functionals on S .
There are now two topologies on S , the metric topologyinduced by the norm and the weak topology induced by F .
The terminology is consistent here since it is possible to showthat the weak topology in fact is weaker than then normtopology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that one writes
X1 × · · · × Xn = {(x1, . . . , xn) | xi ∈ Xi , } .
The projections πi : X1 × · · · × Xn → Xi are given byπi ((x1, . . . , xn)) = xi .
If the space involved have topologies, then the weak topologyassociated with the projections is known as the producttopology on X1 × · · · × Xn.
A net in X1 × · · · × Xn converges if and only if it convergescoordinate-wise.
If Xi are connected, then X1 × · · · × Xn is connected.
If Xi are compact, then X1 × · · · × Xn is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that one writes
X1 × · · · × Xn = {(x1, . . . , xn) | xi ∈ Xi , } .
The projections πi : X1 × · · · × Xn → Xi are given byπi ((x1, . . . , xn)) = xi .
If the space involved have topologies, then the weak topologyassociated with the projections is known as the producttopology on X1 × · · · × Xn.
A net in X1 × · · · × Xn converges if and only if it convergescoordinate-wise.
If Xi are connected, then X1 × · · · × Xn is connected.
If Xi are compact, then X1 × · · · × Xn is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that one writes
X1 × · · · × Xn = {(x1, . . . , xn) | xi ∈ Xi , } .
The projections πi : X1 × · · · × Xn → Xi are given byπi ((x1, . . . , xn)) = xi .
If the space involved have topologies, then the weak topologyassociated with the projections is known as the producttopology on X1 × · · · × Xn.
A net in X1 × · · · × Xn converges if and only if it convergescoordinate-wise.
If Xi are connected, then X1 × · · · × Xn is connected.
If Xi are compact, then X1 × · · · × Xn is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that one writes
X1 × · · · × Xn = {(x1, . . . , xn) | xi ∈ Xi , } .
The projections πi : X1 × · · · × Xn → Xi are given byπi ((x1, . . . , xn)) = xi .
If the space involved have topologies, then the weak topologyassociated with the projections is known as the producttopology on X1 × · · · × Xn.
A net in X1 × · · · × Xn converges if and only if it convergescoordinate-wise.
If Xi are connected, then X1 × · · · × Xn is connected.
If Xi are compact, then X1 × · · · × Xn is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that one writes
X1 × · · · × Xn = {(x1, . . . , xn) | xi ∈ Xi , } .
The projections πi : X1 × · · · × Xn → Xi are given byπi ((x1, . . . , xn)) = xi .
If the space involved have topologies, then the weak topologyassociated with the projections is known as the producttopology on X1 × · · · × Xn.
A net in X1 × · · · × Xn converges if and only if it convergescoordinate-wise.
If Xi are connected, then X1 × · · · × Xn is connected.
If Xi are compact, then X1 × · · · × Xn is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Recall that one writes
X1 × · · · × Xn = {(x1, . . . , xn) | xi ∈ Xi , } .
The projections πi : X1 × · · · × Xn → Xi are given byπi ((x1, . . . , xn)) = xi .
If the space involved have topologies, then the weak topologyassociated with the projections is known as the producttopology on X1 × · · · × Xn.
A net in X1 × · · · × Xn converges if and only if it convergescoordinate-wise.
If Xi are connected, then X1 × · · · × Xn is connected.
If Xi are compact, then X1 × · · · × Xn is compact.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
If Ui is open in Xi , then π−1i (Ui ) is open in X1 × · · · × Xn.
The collection of all such open sets using all the projections isa subbasis for the product topology.
Recall that the open intervals form a basis for the standardtopology of R.
An open rectangle in R2 is produced byπ−1
1 ((a, b)) ∩ π−12 ((c , d)).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
If Ui is open in Xi , then π−1i (Ui ) is open in X1 × · · · × Xn.
The collection of all such open sets using all the projections isa subbasis for the product topology.
Recall that the open intervals form a basis for the standardtopology of R.
An open rectangle in R2 is produced byπ−1
1 ((a, b)) ∩ π−12 ((c , d)).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
If Ui is open in Xi , then π−1i (Ui ) is open in X1 × · · · × Xn.
The collection of all such open sets using all the projections isa subbasis for the product topology.
Recall that the open intervals form a basis for the standardtopology of R.
An open rectangle in R2 is produced byπ−1
1 ((a, b)) ∩ π−12 ((c , d)).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
If Ui is open in Xi , then π−1i (Ui ) is open in X1 × · · · × Xn.
The collection of all such open sets using all the projections isa subbasis for the product topology.
Recall that the open intervals form a basis for the standardtopology of R.
An open rectangle in R2 is produced byπ−1
1 ((a, b)) ∩ π−12 ((c , d)).
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The Cartesian product is not restricted to a finite collection ofsets.
Is is common to work with a product of a countable number ofsets.
It is also possible to work with uncountable products.
Let X = {f : [0, 1]→ R | f continuous} and consider thecollection of functions ϕt : X → R given by ϕt(f ) = f (t).
Let τ be the intersection of all topologies on X so that ϕt iscontinuous for each t ∈ [0, 1].
The topology τ is not the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The Cartesian product is not restricted to a finite collection ofsets.
Is is common to work with a product of a countable number ofsets.
It is also possible to work with uncountable products.
Let X = {f : [0, 1]→ R | f continuous} and consider thecollection of functions ϕt : X → R given by ϕt(f ) = f (t).
Let τ be the intersection of all topologies on X so that ϕt iscontinuous for each t ∈ [0, 1].
The topology τ is not the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The Cartesian product is not restricted to a finite collection ofsets.
Is is common to work with a product of a countable number ofsets.
It is also possible to work with uncountable products.
Let X = {f : [0, 1]→ R | f continuous} and consider thecollection of functions ϕt : X → R given by ϕt(f ) = f (t).
Let τ be the intersection of all topologies on X so that ϕt iscontinuous for each t ∈ [0, 1].
The topology τ is not the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The Cartesian product is not restricted to a finite collection ofsets.
Is is common to work with a product of a countable number ofsets.
It is also possible to work with uncountable products.
Let X = {f : [0, 1]→ R | f continuous} and consider thecollection of functions ϕt : X → R given by ϕt(f ) = f (t).
Let τ be the intersection of all topologies on X so that ϕt iscontinuous for each t ∈ [0, 1].
The topology τ is not the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The Cartesian product is not restricted to a finite collection ofsets.
Is is common to work with a product of a countable number ofsets.
It is also possible to work with uncountable products.
Let X = {f : [0, 1]→ R | f continuous} and consider thecollection of functions ϕt : X → R given by ϕt(f ) = f (t).
Let τ be the intersection of all topologies on X so that ϕt iscontinuous for each t ∈ [0, 1].
The topology τ is not the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
The Cartesian product is not restricted to a finite collection ofsets.
Is is common to work with a product of a countable number ofsets.
It is also possible to work with uncountable products.
Let X = {f : [0, 1]→ R | f continuous} and consider thecollection of functions ϕt : X → R given by ϕt(f ) = f (t).
Let τ be the intersection of all topologies on X so that ϕt iscontinuous for each t ∈ [0, 1].
The topology τ is not the discrete topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Since X is not finite it is possible that τ is metrizablenonetheless.
It turns out that (X , τ) is in fact not metrizable.
This produces a natural example of a topology that cannot bedefined by a metric.
Now consider the countable case and the spaceX = {f : N→ R}.
Think of this X as the space of sequences of real numbers.
Let ϕk : X → R be given by ϕk(f ) = f (k).
The ϕk is sometimes known as the projection onto the k’thcoordinate.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Since X is not finite it is possible that τ is metrizablenonetheless.
It turns out that (X , τ) is in fact not metrizable.
This produces a natural example of a topology that cannot bedefined by a metric.
Now consider the countable case and the spaceX = {f : N→ R}.
Think of this X as the space of sequences of real numbers.
Let ϕk : X → R be given by ϕk(f ) = f (k).
The ϕk is sometimes known as the projection onto the k’thcoordinate.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Since X is not finite it is possible that τ is metrizablenonetheless.
It turns out that (X , τ) is in fact not metrizable.
This produces a natural example of a topology that cannot bedefined by a metric.
Now consider the countable case and the spaceX = {f : N→ R}.
Think of this X as the space of sequences of real numbers.
Let ϕk : X → R be given by ϕk(f ) = f (k).
The ϕk is sometimes known as the projection onto the k’thcoordinate.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Since X is not finite it is possible that τ is metrizablenonetheless.
It turns out that (X , τ) is in fact not metrizable.
This produces a natural example of a topology that cannot bedefined by a metric.
Now consider the countable case and the spaceX = {f : N→ R}.
Think of this X as the space of sequences of real numbers.
Let ϕk : X → R be given by ϕk(f ) = f (k).
The ϕk is sometimes known as the projection onto the k’thcoordinate.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Since X is not finite it is possible that τ is metrizablenonetheless.
It turns out that (X , τ) is in fact not metrizable.
This produces a natural example of a topology that cannot bedefined by a metric.
Now consider the countable case and the spaceX = {f : N→ R}.
Think of this X as the space of sequences of real numbers.
Let ϕk : X → R be given by ϕk(f ) = f (k).
The ϕk is sometimes known as the projection onto the k’thcoordinate.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Since X is not finite it is possible that τ is metrizablenonetheless.
It turns out that (X , τ) is in fact not metrizable.
This produces a natural example of a topology that cannot bedefined by a metric.
Now consider the countable case and the spaceX = {f : N→ R}.
Think of this X as the space of sequences of real numbers.
Let ϕk : X → R be given by ϕk(f ) = f (k).
The ϕk is sometimes known as the projection onto the k’thcoordinate.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Since X is not finite it is possible that τ is metrizablenonetheless.
It turns out that (X , τ) is in fact not metrizable.
This produces a natural example of a topology that cannot bedefined by a metric.
Now consider the countable case and the spaceX = {f : N→ R}.
Think of this X as the space of sequences of real numbers.
Let ϕk : X → R be given by ϕk(f ) = f (k).
The ϕk is sometimes known as the projection onto the k’thcoordinate.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let
d(f , g) =∞∑k=1
min{|f (k)− g(k)|, 1}2k
.
It requires quite a bit of work to check that this indeed is ametric and that the metric topology is in fact the weaktopology, here also known as the product topology.
The use of the intersection of family of topologies, where eachmember is somewhat mysterious, creates an object that isdifficult to understand.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let
d(f , g) =∞∑k=1
min{|f (k)− g(k)|, 1}2k
.
It requires quite a bit of work to check that this indeed is ametric and that the metric topology is in fact the weaktopology, here also known as the product topology.
The use of the intersection of family of topologies, where eachmember is somewhat mysterious, creates an object that isdifficult to understand.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let
d(f , g) =∞∑k=1
min{|f (k)− g(k)|, 1}2k
.
It requires quite a bit of work to check that this indeed is ametric and that the metric topology is in fact the weaktopology, here also known as the product topology.
The use of the intersection of family of topologies, where eachmember is somewhat mysterious, creates an object that isdifficult to understand.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
For a general family ϕα : X → Xα various sets ϕ−1α (Uα), where
Uα is open in Xα, form a subbasis for the weak topology.
By restricting the Uα to a basis or even a subbasis in Xα it ispossible to get a better understanding of the weak topology.
Consider the case R2 and the vertical open strips of the form{(x , y) ∈ R2 | a < x < b & y ∈ R
},
and their horizontal counter parts.
Now imagine finite intersections and their unions to visualizethe open sets in the product topology.
It is of course necessary to use infinitely many strips to createan open disk.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
For a general family ϕα : X → Xα various sets ϕ−1α (Uα), where
Uα is open in Xα, form a subbasis for the weak topology.
By restricting the Uα to a basis or even a subbasis in Xα it ispossible to get a better understanding of the weak topology.
Consider the case R2 and the vertical open strips of the form{(x , y) ∈ R2 | a < x < b & y ∈ R
},
and their horizontal counter parts.
Now imagine finite intersections and their unions to visualizethe open sets in the product topology.
It is of course necessary to use infinitely many strips to createan open disk.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
For a general family ϕα : X → Xα various sets ϕ−1α (Uα), where
Uα is open in Xα, form a subbasis for the weak topology.
By restricting the Uα to a basis or even a subbasis in Xα it ispossible to get a better understanding of the weak topology.
Consider the case R2 and the vertical open strips of the form{(x , y) ∈ R2 | a < x < b & y ∈ R
},
and their horizontal counter parts.
Now imagine finite intersections and their unions to visualizethe open sets in the product topology.
It is of course necessary to use infinitely many strips to createan open disk.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
For a general family ϕα : X → Xα various sets ϕ−1α (Uα), where
Uα is open in Xα, form a subbasis for the weak topology.
By restricting the Uα to a basis or even a subbasis in Xα it ispossible to get a better understanding of the weak topology.
Consider the case R2 and the vertical open strips of the form{(x , y) ∈ R2 | a < x < b & y ∈ R
},
and their horizontal counter parts.
Now imagine finite intersections and their unions to visualizethe open sets in the product topology.
It is of course necessary to use infinitely many strips to createan open disk.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
For a general family ϕα : X → Xα various sets ϕ−1α (Uα), where
Uα is open in Xα, form a subbasis for the weak topology.
By restricting the Uα to a basis or even a subbasis in Xα it ispossible to get a better understanding of the weak topology.
Consider the case R2 and the vertical open strips of the form{(x , y) ∈ R2 | a < x < b & y ∈ R
},
and their horizontal counter parts.
Now imagine finite intersections and their unions to visualizethe open sets in the product topology.
It is of course necessary to use infinitely many strips to createan open disk.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let ∼ be an equivalence relation on a set X .
Denote the equivalence classes by X/ ∼.
Let π : X → X/ ∼ be the map π(x) = [x ], where [x ] denotesthe equivalence class that contains x .
Suppose X has a topology τ .
Declare U ⊂ X/ ∼ open if and only if π−1(U) ∈ τ .
The inverse image preserves both unions and intersections sothis indeed a topology known as the identification topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let ∼ be an equivalence relation on a set X .
Denote the equivalence classes by X/ ∼.
Let π : X → X/ ∼ be the map π(x) = [x ], where [x ] denotesthe equivalence class that contains x .
Suppose X has a topology τ .
Declare U ⊂ X/ ∼ open if and only if π−1(U) ∈ τ .
The inverse image preserves both unions and intersections sothis indeed a topology known as the identification topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let ∼ be an equivalence relation on a set X .
Denote the equivalence classes by X/ ∼.
Let π : X → X/ ∼ be the map π(x) = [x ], where [x ] denotesthe equivalence class that contains x .
Suppose X has a topology τ .
Declare U ⊂ X/ ∼ open if and only if π−1(U) ∈ τ .
The inverse image preserves both unions and intersections sothis indeed a topology known as the identification topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let ∼ be an equivalence relation on a set X .
Denote the equivalence classes by X/ ∼.
Let π : X → X/ ∼ be the map π(x) = [x ], where [x ] denotesthe equivalence class that contains x .
Suppose X has a topology τ .
Declare U ⊂ X/ ∼ open if and only if π−1(U) ∈ τ .
The inverse image preserves both unions and intersections sothis indeed a topology known as the identification topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let ∼ be an equivalence relation on a set X .
Denote the equivalence classes by X/ ∼.
Let π : X → X/ ∼ be the map π(x) = [x ], where [x ] denotesthe equivalence class that contains x .
Suppose X has a topology τ .
Declare U ⊂ X/ ∼ open if and only if π−1(U) ∈ τ .
The inverse image preserves both unions and intersections sothis indeed a topology known as the identification topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let ∼ be an equivalence relation on a set X .
Denote the equivalence classes by X/ ∼.
Let π : X → X/ ∼ be the map π(x) = [x ], where [x ] denotesthe equivalence class that contains x .
Suppose X has a topology τ .
Declare U ⊂ X/ ∼ open if and only if π−1(U) ∈ τ .
The inverse image preserves both unions and intersections sothis indeed a topology known as the identification topology.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A common example is to ‘collapse a set to a point’.
Suppose A ⊂ X and partition X into singletons {x}, whenx /∈ A, together with the set A itself.
The associated equivalence relation produces a space X/ ∼that frequently is written X/A.
Let X = [0, 1] and A = {0, 1}.
Let φ : X/A→ S1 be given by φ([x ]) = e2πix .
The map is well-defined since φ([0]) = e2πi0 = 1 andφ([1]) = e2πi = 1.
The map is in fact a set-isomorphism.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A common example is to ‘collapse a set to a point’.
Suppose A ⊂ X and partition X into singletons {x}, whenx /∈ A, together with the set A itself.
The associated equivalence relation produces a space X/ ∼that frequently is written X/A.
Let X = [0, 1] and A = {0, 1}.
Let φ : X/A→ S1 be given by φ([x ]) = e2πix .
The map is well-defined since φ([0]) = e2πi0 = 1 andφ([1]) = e2πi = 1.
The map is in fact a set-isomorphism.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A common example is to ‘collapse a set to a point’.
Suppose A ⊂ X and partition X into singletons {x}, whenx /∈ A, together with the set A itself.
The associated equivalence relation produces a space X/ ∼that frequently is written X/A.
Let X = [0, 1] and A = {0, 1}.
Let φ : X/A→ S1 be given by φ([x ]) = e2πix .
The map is well-defined since φ([0]) = e2πi0 = 1 andφ([1]) = e2πi = 1.
The map is in fact a set-isomorphism.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A common example is to ‘collapse a set to a point’.
Suppose A ⊂ X and partition X into singletons {x}, whenx /∈ A, together with the set A itself.
The associated equivalence relation produces a space X/ ∼that frequently is written X/A.
Let X = [0, 1] and A = {0, 1}.
Let φ : X/A→ S1 be given by φ([x ]) = e2πix .
The map is well-defined since φ([0]) = e2πi0 = 1 andφ([1]) = e2πi = 1.
The map is in fact a set-isomorphism.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A common example is to ‘collapse a set to a point’.
Suppose A ⊂ X and partition X into singletons {x}, whenx /∈ A, together with the set A itself.
The associated equivalence relation produces a space X/ ∼that frequently is written X/A.
Let X = [0, 1] and A = {0, 1}.
Let φ : X/A→ S1 be given by φ([x ]) = e2πix .
The map is well-defined since φ([0]) = e2πi0 = 1 andφ([1]) = e2πi = 1.
The map is in fact a set-isomorphism.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A common example is to ‘collapse a set to a point’.
Suppose A ⊂ X and partition X into singletons {x}, whenx /∈ A, together with the set A itself.
The associated equivalence relation produces a space X/ ∼that frequently is written X/A.
Let X = [0, 1] and A = {0, 1}.
Let φ : X/A→ S1 be given by φ([x ]) = e2πix .
The map is well-defined since φ([0]) = e2πi0 = 1 andφ([1]) = e2πi = 1.
The map is in fact a set-isomorphism.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
A common example is to ‘collapse a set to a point’.
Suppose A ⊂ X and partition X into singletons {x}, whenx /∈ A, together with the set A itself.
The associated equivalence relation produces a space X/ ∼that frequently is written X/A.
Let X = [0, 1] and A = {0, 1}.
Let φ : X/A→ S1 be given by φ([x ]) = e2πix .
The map is well-defined since φ([0]) = e2πi0 = 1 andφ([1]) = e2πi = 1.
The map is in fact a set-isomorphism.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose [xλ]→ [x∞] in X/A.
If [x∞] 6= [0], then [x∞] = [x ] with x ∈ (0, 1), and
φ([xλ]) = e2πixλ → e2πix∞ = φ([x∞]).
If [x∞] = [0], then it must be understood that the relevantopen sets have the form U = [0, β) ∪ (α, 1] and not only one ofthe two sets in the union.
Keeping this in mind, it is seen that φ([xλ])→ 1 = φ([x∞]).
It follows that φ is continuous.
Now look at φ−1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose [xλ]→ [x∞] in X/A.
If [x∞] 6= [0], then [x∞] = [x ] with x ∈ (0, 1), and
φ([xλ]) = e2πixλ → e2πix∞ = φ([x∞]).
If [x∞] = [0], then it must be understood that the relevantopen sets have the form U = [0, β) ∪ (α, 1] and not only one ofthe two sets in the union.
Keeping this in mind, it is seen that φ([xλ])→ 1 = φ([x∞]).
It follows that φ is continuous.
Now look at φ−1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose [xλ]→ [x∞] in X/A.
If [x∞] 6= [0], then [x∞] = [x ] with x ∈ (0, 1), and
φ([xλ]) = e2πixλ → e2πix∞ = φ([x∞]).
If [x∞] = [0], then it must be understood that the relevantopen sets have the form U = [0, β) ∪ (α, 1] and not only one ofthe two sets in the union.
Keeping this in mind, it is seen that φ([xλ])→ 1 = φ([x∞]).
It follows that φ is continuous.
Now look at φ−1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose [xλ]→ [x∞] in X/A.
If [x∞] 6= [0], then [x∞] = [x ] with x ∈ (0, 1), and
φ([xλ]) = e2πixλ → e2πix∞ = φ([x∞]).
If [x∞] = [0], then it must be understood that the relevantopen sets have the form U = [0, β) ∪ (α, 1] and not only one ofthe two sets in the union.
Keeping this in mind, it is seen that φ([xλ])→ 1 = φ([x∞]).
It follows that φ is continuous.
Now look at φ−1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose [xλ]→ [x∞] in X/A.
If [x∞] 6= [0], then [x∞] = [x ] with x ∈ (0, 1), and
φ([xλ]) = e2πixλ → e2πix∞ = φ([x∞]).
If [x∞] = [0], then it must be understood that the relevantopen sets have the form U = [0, β) ∪ (α, 1] and not only one ofthe two sets in the union.
Keeping this in mind, it is seen that φ([xλ])→ 1 = φ([x∞]).
It follows that φ is continuous.
Now look at φ−1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Suppose [xλ]→ [x∞] in X/A.
If [x∞] 6= [0], then [x∞] = [x ] with x ∈ (0, 1), and
φ([xλ]) = e2πixλ → e2πix∞ = φ([x∞]).
If [x∞] = [0], then it must be understood that the relevantopen sets have the form U = [0, β) ∪ (α, 1] and not only one ofthe two sets in the union.
Keeping this in mind, it is seen that φ([xλ])→ 1 = φ([x∞]).
It follows that φ is continuous.
Now look at φ−1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let zλ ∈ S1 be such that zλ → z∞.
If z∞ 6= 1, then eventually zλ = e2πixλ with xλ ∈ (0, 1).
Moreover, xλ → x∞ where z∞ = e2πix∞ .
Finally, if z∞ = 1 then eventually zλ = e2πixλ has xλ ∈ U nomatter what the feasible choice is of 0 < β < α < 1.
This implies that xλ eventually is in any open set [U]containing [0].
So φ1(zλ)→ φ−1(z∞) in either case, and hence φ−1 iscontinuous.
Putting things together, [0, 1]/{0, 1} is homeomorphic to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let zλ ∈ S1 be such that zλ → z∞.
If z∞ 6= 1, then eventually zλ = e2πixλ with xλ ∈ (0, 1).
Moreover, xλ → x∞ where z∞ = e2πix∞ .
Finally, if z∞ = 1 then eventually zλ = e2πixλ has xλ ∈ U nomatter what the feasible choice is of 0 < β < α < 1.
This implies that xλ eventually is in any open set [U]containing [0].
So φ1(zλ)→ φ−1(z∞) in either case, and hence φ−1 iscontinuous.
Putting things together, [0, 1]/{0, 1} is homeomorphic to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let zλ ∈ S1 be such that zλ → z∞.
If z∞ 6= 1, then eventually zλ = e2πixλ with xλ ∈ (0, 1).
Moreover, xλ → x∞ where z∞ = e2πix∞ .
Finally, if z∞ = 1 then eventually zλ = e2πixλ has xλ ∈ U nomatter what the feasible choice is of 0 < β < α < 1.
This implies that xλ eventually is in any open set [U]containing [0].
So φ1(zλ)→ φ−1(z∞) in either case, and hence φ−1 iscontinuous.
Putting things together, [0, 1]/{0, 1} is homeomorphic to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let zλ ∈ S1 be such that zλ → z∞.
If z∞ 6= 1, then eventually zλ = e2πixλ with xλ ∈ (0, 1).
Moreover, xλ → x∞ where z∞ = e2πix∞ .
Finally, if z∞ = 1 then eventually zλ = e2πixλ has xλ ∈ U nomatter what the feasible choice is of 0 < β < α < 1.
This implies that xλ eventually is in any open set [U]containing [0].
So φ1(zλ)→ φ−1(z∞) in either case, and hence φ−1 iscontinuous.
Putting things together, [0, 1]/{0, 1} is homeomorphic to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let zλ ∈ S1 be such that zλ → z∞.
If z∞ 6= 1, then eventually zλ = e2πixλ with xλ ∈ (0, 1).
Moreover, xλ → x∞ where z∞ = e2πix∞ .
Finally, if z∞ = 1 then eventually zλ = e2πixλ has xλ ∈ U nomatter what the feasible choice is of 0 < β < α < 1.
This implies that xλ eventually is in any open set [U]containing [0].
So φ1(zλ)→ φ−1(z∞) in either case, and hence φ−1 iscontinuous.
Putting things together, [0, 1]/{0, 1} is homeomorphic to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let zλ ∈ S1 be such that zλ → z∞.
If z∞ 6= 1, then eventually zλ = e2πixλ with xλ ∈ (0, 1).
Moreover, xλ → x∞ where z∞ = e2πix∞ .
Finally, if z∞ = 1 then eventually zλ = e2πixλ has xλ ∈ U nomatter what the feasible choice is of 0 < β < α < 1.
This implies that xλ eventually is in any open set [U]containing [0].
So φ1(zλ)→ φ−1(z∞) in either case, and hence φ−1 iscontinuous.
Putting things together, [0, 1]/{0, 1} is homeomorphic to S1.
Topology
Linner
Evidence oftopology.
The logarithm.
Existence?
An exampleinvolving thelogarithm.
Work and flux.
Potential andcurl.
Potential andcurl.
Flux withoutdivergence?
Detectingdifferences.
How do thedomains differ?
The point withanother point.
Adding a pointto R2.
If not from ametric, whatis topology?
Open intervals.
Open sets in R
Open sets inmetric spaces
Topology.
Essentially thesametopologies.
Homeomorphisms.
Closed sets.
Closed sets.
Counting finitetopologies.
Separation.
Connectedspaces.
Closures.
Interiors.
Interiors.
Closures andcomplements.
Compactspaces.
Maps.
Let zλ ∈ S1 be such that zλ → z∞.
If z∞ 6= 1, then eventually zλ = e2πixλ with xλ ∈ (0, 1).
Moreover, xλ → x∞ where z∞ = e2πix∞ .
Finally, if z∞ = 1 then eventually zλ = e2πixλ has xλ ∈ U nomatter what the feasible choice is of 0 < β < α < 1.
This implies that xλ eventually is in any open set [U]containing [0].
So φ1(zλ)→ φ−1(z∞) in either case, and hence φ−1 iscontinuous.
Putting things together, [0, 1]/{0, 1} is homeomorphic to S1.
