This document discusses extensions of mappings from subspaces into topologically complete spaces. It begins with terminology for topological concepts like zero-dimensional, normal, and Dieudonne complete spaces. It then presents several theorems about extending discrete-valued and continuous mappings from subspaces into metric and Banach spaces if the closure of the subspace in the completion is equal to the completion of the subspace. The document concludes with a bibliography of related works.

1. About extensions of mappings into
topologically complete spaces
Radu Dumbraveanu
Alecu Russo Balti State University
IMCS-50
August 19-23, 2014
Chisinau, Republic of Moldova
R. Dumbraveanu Extensions of mappings into topologically complete spaces

2. 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 small
inductive dimension zero (indX = 0), i.e. X has a base of clopen
sets.
A normal space X has large inductive dimension zero (IndX = 0) if
and only if for any two disjoint closed subsets A and B of X there
is a clopen set C such that A C and B (X n C).
A normal space X has Lebesgue covering dimension zero
(dimX = 0) if any

3. nite open cover of X can be re

4. ned to a
partition of X into clopen sets.
R. Dumbraveanu Extensions of mappings into topologically complete spaces

5. 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.
R. Dumbraveanu Extensions of mappings into topologically complete spaces

6. Terminology
A topological space X is Dieudonne complete if there exists a
complete uniformity on the space X.
A space X is topologically complete if X is homeomorphic to a
closed subspace of a product of metrizable spaces.
The Dieudonne completion X of a space X is a topological
complete space for which X is a dense subspace of X and each
continuous mapping g from X into a topologically complete space
Y admits a continuous extension g over X.
A family fF : 2 Ag of the space X is functionally discrete if
there exists a family ff : 2 Ag of continuous functions on X
such that the family ff 1
(0; 1) : 2 Ag is discrete in X and
F f 1
(1) for each 2 Ag.
R. Dumbraveanu Extensions of mappings into topologically complete spaces

7. On extension of discrete-valued mappings
Theorem (1.1)
Let Y X, X normal and dimX = 0, then the following assertions
are equivalent:
1 For every clopen subset U of Y the set clXU is clopen in
clXY .
2 For every clopen partition
= fU;Vg of Y there exists a
clopen partition
0 = fU0;V0g of X such that U = U0 Y
and V = V0 Y .
3 Every function f 2 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.
R. Dumbraveanu Extensions of mappings into topologically complete spaces

8. On extension of discrete-valued mappings
Example
Let Y = N with the discrete topology and X =

9. N. Then X is
normal and dimX = 0. Let !. Then every continuous function
from Y into D2 extends to a continuous function on X. But if
! then, since a continuous function on a compact space must
be bounded, not every continuous function from Y into D
extends to a continuous function on X. Thus, in case of
continuous functions into a in

10. nite discrete space, the conditions
for X to be normal and dimX = 0 are not enough.
R. Dumbraveanu Extensions of mappings into topologically complete spaces

11. On extension of discrete-valued mappings
Theorem (1.2)
Let Y X, X be a collectionwise normal space and dimX = 0.
Then the following are equivalent:
1 For every cardinal and a discrete collection fU : 2 D g
of clopen subsets of Y the collection fclXU : 2 D g is
discrete in X.
2 For every clopen subset U of Y the set clXU is clopen in
clXY and every discrete collection fU : 2 Ag of clopen
subsets of Y is locally

12. nite in X.
3 For each discrete space Z every function f 2 C(Y ; Z) extends
to a function in C(X; Z).
4 If Z is a topologically complete space and f 2 C(Y ; Z), then
there exists g 2 C(clXY ; Z) such that f = gjY .
5 clXY Y .
R. Dumbraveanu Extensions of mappings into topologically complete spaces

13. Extension of mappings into metric spaces
Theorem (2.1)
Let Y be a subspace of the space X, E be a topologically complete
space and for each closed subspace Z of X and any continuous
mapping g : Z ! E there exists a continuous extension
g : X ! E. If Y = clXY , then for each continuous mapping
g : Y ! E there exists a continuous extension g : X ! E.
R. Dumbraveanu Extensions of mappings into topologically complete spaces

14. Extension of mappings into metric spaces
Theorem (2.2)
Let Y be a subspace of the space X, and for any continuous
mapping g : Z ! E of a closed subspace Z of X into a Banach
space E there exists a continuous extension g : X ! E. Then the
following assertions are equivalent:
1 Y = clXY ,
2 For each continuous mapping g : Y ! E into a Banach
space E there exists a continuous extension g : X ! E.
3 For each continuous mapping g : Y ! E into a metrizable
space E there exists a continuous extension g : clXY ! E.
4 For each functionally discrete family fF : 2 Ag of the
space Y the family fclX F : 2 Ag is discrete in X.
R. Dumbraveanu Extensions of mappings into topologically complete spaces

15. Bibliography
R. L. Ellis, Extending continuous functions on
zero-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-valued
functions, Transactions of AMS, 324, 1 (1971), 293-302.
S. Nedev, Selected theorems on multivalued sections and
extensions (in Russian), Serdica, 1 (1975), 285-294.
R. Dumbraveanu Extensions of mappings into topologically complete spaces