About extensions of mappings into topologically complete spaces

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