About extensions of mappings into topologically complete spaces

About extensions of mappings intotopologically complete spaces

Radu Dumbraveanu

Alecu Russo Balti State University

IMCS-50August 19-23, 2014

Chisinau, Republic of Moldova

R. Dumbraveanu Extensions of mappings into topologically complete spaces

Terminology

Every space is considered to be a completely regular T1-space.

A regular space X is said to be zero-dimensional if it is of smallinductive dimension zero (indX = 0), i.e. X has a base of clopensets.

A normal space X has large inductive dimension zero (IndX = 0) ifand only if for any two disjoint closed subsets A and B of X thereis a clopen set C such that A ⊆ C and B ⊆ (X \ C ).

A normal space X has Lebesgue covering dimension zero(dimX = 0) if any finite open cover of X can be refined to apartition of X into clopen sets.


Terminology

It is well known that:

for any metric space X , IndX = dimX ;

if X is Lindelof then indX = 0 if and only if IndX = 0;

if X is normal then IndX = 0 if and only if dimX = 0.


Terminology

A topological space X is Dieudonne complete if there exists acomplete uniformity on the space X .

A space X is topologically complete if X is homeomorphic to aclosed subspace of a product of metrizable spaces.

The Dieudonne completion µX of a space X is a topologicalcomplete space for which X is a dense subspace of µX and eachcontinuous mapping g from X into a topologically complete spaceY admits a continuous extension µg over µX .

A family {Fα : α ∈ A} of the space X is functionally discrete ifthere exists a family {fα : α ∈ A} of continuous functions on Xsuch that the family {f −1α (0, 1) : α ∈ A} is discrete in X andFα ⊆ f −1α (1) for each α ∈ A}.


On extension of discrete-valued mappings

Theorem (1.1)

Let Y ⊆ X , X normal and dimX = 0, then the following assertionsare equivalent:

1 For every clopen subset U of Y the set clXU is clopen inclXY .

2 For every clopen partition γ = {U,V } of Y there exists aclopen partition γ′ = {U ′,V ′} of X such that U = U ′ ∩ Yand V = V ′ ∩ Y .

3 Every function f ∈ C (Y ,D) extends to a function in C (X ,D).

Remark

The equivalence 1↔2 from Theorem 1.1 is true for any space X .



Example

Let Y = N with the discrete topology and X = βN. Then X isnormal and dimX = 0. Let τ < ω. Then every continuous functionfrom Y into D2 extends to a continuous function on X . But ifτ ≥ ω then, since a continuous function on a compact space mustbe bounded, not every continuous function from Y into Dτ

extends to a continuous function on X . Thus, in case ofcontinuous functions into a infinite discrete space, the conditionsfor X to be normal and dimX = 0 are not enough.



Theorem (1.2)

Let Y ⊆ X , X be a collectionwise normal space and dimX = 0.Then the following are equivalent:

1 For every cardinal τ and a discrete collection {Uα : α ∈ Dτ}of clopen subsets of Y the collection {clXUα : α ∈ Dτ} isdiscrete in X .

2 For every clopen subset U of Y the set clXU is clopen inclXY and every discrete collection {Uα : α ∈ A} of clopensubsets of Y is locally finite in X .

3 For each discrete space Z every function f ∈ C (Y ,Z ) extendsto a function in C (X ,Z ).

4 If Z is a topologically complete space and f ∈ C (Y ,Z ), thenthere exists g ∈ C (clXY ,Z ) such that f = g |Y .

5 clµXY ⊆ µY .


Extension of mappings into metric spaces

Theorem (2.1)

Let Y be a subspace of the space X , E be a topologically completespace and for each closed subspace Z of X and any continuousmapping g : Z −→ E there exists a continuous extensiong : X −→ E . If µY = clµXY , then for each continuous mappingg : Y −→ E there exists a continuous extension g : X −→ E .


Extension of mappings into metric spaces

Theorem (2.2)

Let Y be a subspace of the space X , and for any continuousmapping g : Z −→ E of a closed subspace Z of X into a Banachspace E there exists a continuous extension g : X −→ E . Then thefollowing assertions are equivalent:

1 µY = clµXY ,

2 For each continuous mapping g : Y −→ E into a Banachspace E there exists a continuous extension g : X −→ E .

3 For each continuous mapping g : Y −→ E into a metrizablespace E there exists a continuous extension µg : clXY −→ E .

4 For each functionally discrete family {Fα : α ∈ A} of thespace Y the family {clXFα : α ∈ A} is discrete in X .


Bibliography

R. L. Ellis, Extending continuous functions onzero-dimensional spaces, Math. Annal. 186 (1970), 114-122.

R. Engelking, Dimension theory, North-Holland, Amsterdam,1978.

R. Engelking, General Topology, PWN, Warszawa, 1977.

J. Kulesza, R. Levy, P. Nyikos, Extending discrete-valuedfunctions, Transactions of AMS, 324, 1 (1971), 293-302.

S. Nedev, Selected theorems on multivalued sections andextensions (in Russian), Serdica, 1 (1975), 285-294.


About extensions of mappings into topologically complete spaces

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