INTRODUCTION TO COMPUTATIONAL...

INTRODUCTION TOCOMPUTATIONAL TOPOLOGY

Afra Zomorodian

CS 468 – Lecture 1

1-14-4

Afra Zomorodian – CS 468 Lecture 1 -

ORGANIZATION

• Wednesdays, 12:30-2 PM, in Gates 392

• Lectures

• Papers

• Projects


WHY ORGANIZED?

• Topological problems arise in computer science

• We don’t know topology

• Topology is

– large

– unfamiliar

– axiomatic (therefore unintuitive)

– cryptic

• Goal: present background for computer scientists

• So, you can read papers!


WHAT IS TOPOLOGY?

• Not how things look (geometry)

• But how they areconnected

• Classifications

• Invariants

1. transform space in a fixed way

2. observe what stays the same

• Erlanger Programm (Felix Klein)

• Intrinsic vs. Extrinsic


MOTIVATIONGRAPHICS: SURFACE RECONSTRUCTION


MOTIVATIONGRAPHICS: TUNNELS


MOTIVATIONROBOTICS: CONFIGURATION SPACE

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MOTIVATIONGEOGRAPHY: TERRAINS


MOTIVATIONBIOLOGY: STRUCTURE


MOTIVATIONCHEMISTRY: KNOTS & L INKS


POINT SET TOPOLOGY

Afra Zomorodian

CS 468 – Lecture 1

1-14-4


MOTIVATION

• Connectivity

• Neighborhoods

• f : D ⊆ R → R

• ε-δ definition: limx→x0 f(x) = y0 iff for all ε > 0, ∃δ > 0 such that

if x ∈ D and|x− x0| < δ, then|f(x)− y0| < ε.

• Mapping of open intervals

• Metric


SETS OFPOINTS

• A set is a well-defined collection of objects, such that

1. elementsa ∈ S.

2. oneempty set∅.

3. description:{x | P(x)} or {1, 2, 3}

4. well-definedif a ∈ S or a 6∈ S

• Point


TOPOLOGICAL SPACES

• Given setX

• TakeT ⊆ 2X such that:

1. If S1, S2 ∈ T , thenS1 ∩ S2 ∈ T .

2. If {SJ | j ∈ J} ⊆ T , then∪j∈JSj ∈ T .

3. ∅, X ∈ T .

• T is atopologyon setX

• S ∈ T is anopen set.

• Complement ofS is closed.

• All possibilities

• finite intersections, infinite unions

• The pair(X, T ) topological spaceX


CONTINUITY

fA

YX

−1f (A)

• f : X → Y

• Open setA in Y

• Supposef−1(A) is open inX.

• f is continuous

• f is amap


SET PROPERTIES

A ⊆ X


CLOSURE

(a)A ⊆ X (b) A

TheclosureA of A is the intersection of all closed sets containingA.


INTERIOR

(a)A ⊆ X (b) A

Theinterior A of A is the union of all open sets contained inA.


BOUNDARY

(a)A ⊆ X (b) ∂A

Theboundary∂A of A is ∂A = A− A.


NEIGHBORHOODS

• X = (X, T ), a topological space.

• x ∈ X

• A ∈ T such thatx ∈ A is aneighborhoodof x

• Suppose we have a collectionB of neighborhoods ofx

• Every neighborhood ofx contains a neighborhood inB

• We callB abasis of neighborhoods atx ∈ X


SUBSPACES

• X = (X, T ), a topological space.

• A ⊆ X, a subset

• TA = {S ∩ A | S ∈ T}

• TA is therelativeor inducedtopology

• A = (A, TA) is a topological space, asubspaceof X.

• Not the only topology possible (as we will see)


METRIC

• A metricor distance functiond : X ×X → R is a function that

satisfies:

1. For allx, y ∈ X, d(x, y) ≥ 0 (positivity).

2. If d(x, y) = 0, thenx = y (non-degeneracy).

3. For allx, y ∈ X, d(x, y) = d(y, x) (symmetry).

4. For allx, y, z ∈ X, d(x, y) + d(y, z) ≥ d(x, z) (the triangle

inequality).

• Euclidean metric: d(x, y) =√∑n

i=1(ui(x)− ui(y))2


METRIC SPACES

• Theopen ballB(x, r) with centerx and radiusr > 0 with respect to

metricd is defined to beB(x, r) = {y | d(x, y) < r}.

• A setX with a metric functiond is called ametric space.

• Endowed withmetric topologyof d, where the set of open balls

defined usingd serve as basis neighborhoods.

• The Cartesian product ofn copies ofR along with the Euclidean

metric is then-dimensional Euclidean spaceRn.

• Circle Example


INTUITION

• We loveEuclidean spaces

• Spaces that look like them? Manifolds!

• How aboutlocally Euclidean?

• Map local pieces to Euclidean spaces

• We don’t want the dimension to vary much

• Sphere


HOMEOMORPHISMS

• f : X → Y, 1-1, onto

• f is continuous (a map)

• f−1 is continuous

• f is ahomeomorphism(bijective bicontinuous)

• X is homeomorphicto Y

• X ≈ Y

• X andY have the sametopological type


CHART

p

p’

U

U’

ϕ

ϕ

X

−1

IRd

• p ∈ U ⊆ X

• ϕ : U → U ′ ⊆ Rd is a homeomorphism

• ϕ is achart

• It hasdimensiond

• ui : Rn → R standard coordinates onRd

• xi = ui ◦ ϕ : U → R arecoordinate functionsof ϕ


STRANGE SPACES

• Given two distinct pointsx, y ∈ X, x 6= y

• U , a neighborhood ofx

• V , a neighborhood ofy

• U ∩ V = ∅

• Then,X is Hausdorff.

• X is separableif it has a countable basis of neighborhoods.

• Metric spaces are always Hausdorff and separable (proof)


MANIFOLD

• X is separable, Hausdorff

• Hasd-dimensional chart at every pointx ∈ X (locally like Rd)

• X is a(topological, abstract)d-manifoldwith dimensiond

2-Manifolds


MANIFOLD WITH BOUNDARY

• X is separable, Hausdorff

• d-dimensional chart at most points

• Some points have neighborhoods homeomorphic to

Hd = {x ∈ Rd | x1 ≥ 0}, X is ad-manifold with boundary

• Theboundary∂X of X is the set of points with neighborhood

homeomorphic toHd.

• (Theorem)∂X is a(d− 1)-manifold.


COMPACTNESS

• A covering ofA ⊆ X is a family{Cj | j ∈ J} in 2X , such that

A ⊆⋃

j∈J Cj .

• An open coveringis a covering consisting of open sets.

• A subcoveringof a covering{Cj | j ∈ J} is a covering

{Ck | k ∈ K}, whereK ⊆ J .

• A ⊆ X is compactif every open covering ofA has a finite

subcovering.

. . .

Finite area, not compact


EMBEDDING

• Homeomorphisms allow us to place one manifold within another

• g : X → Y

• g is homeomorphism onto its imageg(X)

• g is anembedding

• g(X) is anembedded submanifold

• We give it the relative topology inY

• We are most familiar with embedded manifolds


NON-EMBEDDING

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F : R → R2

F (t) = 2 cos(t− π/2), sin(2(t− π/2))


A FIX

0

π

2π

-40 -20 0 20 40

g : R → (0, 2π)g(t) = π + 2 tan−1(t)


EMBEDDING?

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F (t) = F (g(t))


IMMERSIONS

• Nasty things can happen for non-compact manifolds

• Definition requires smooth notions

• For compact manifolds, animmersionallows self-intersection

Standard immersion of the Klein bottle


WHAT TO REMEMBER

• Topology worries about connectivity

• A topology is a set of open sets that define neighborhoods

• A manifold is locally Euclidean

• Homeomorphisms map manifolds to each other

• Natural question: which spaces are homeomorphic to each other?


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