2
$\begingroup$

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $g:X \rightarrow \mathbb{R}$.

The proof can be found here. I came across the following proposition:

A space $X$ is normal iff every lower semi-continuous multi-valued map $F: X \rightarrow 2^\mathbb{R}$ with compact and convex images admits a continuous selection.

The statement maybe requires some additional definitions:

  1. A multi-valued map $F: X \rightarrow 2^Y$ is called lower semi-continuous if for every $G \subseteq Y$ open, $\left\{x \in X \mid F(x) \cap G \neq \emptyset\right\}$ is also an open set.
  2. If $F: X \rightarrow 2^Y$ is a lower semi-continuous multi-valued map, we call $f: X \rightarrow Y$ a continuous selection if $f$ is continuous and $f(x) \in F(x)$ holds for all $x \in X$.

I don't know if this is needed to prove the proposition, but I'll include it anyways; I've proved that $c(F)$, defined by $c(F)(x) = c(F(x))$ (where $c(A)$ denotes the convex hull of a set $A$), is lower semi-continuous if $F: X \rightarrow 2^Y$.

Any help with proving the blockquoted statement is appreciated, I don't really know how to get started. It does state however that the proof uses the Tietze extension theorem.

$\endgroup$
3
  • $\begingroup$ Did you look at Ernest Michael's 1956 Annals article? $\endgroup$
    – user1688
    Commented Jun 17, 2016 at 8:46
  • $\begingroup$ In fact, I did. Michael states that the if directions follows directly from the Tietze extension theorem (Theorem 3.1. in the article), but this is not completely clear to me. Also, the iff direction is not proven in the article if I'm not mistaken. $\endgroup$ Commented Jun 17, 2016 at 10:04
  • 1
    $\begingroup$ "Proofwiki" is not reliable. Please, consult Engelking. $\endgroup$ Commented Jun 18, 2016 at 5:19

1 Answer 1

1
$\begingroup$

The following is true

A space X is normal iff every lower semi-continuous multi-valued map $F:X\rightarrow 2^\mathbb R$ such that for every $x\in X$, $F(x)$ is either convex and compact or $F(x)=\mathbb R$, admits a continuous selection.

(See Repovs, Semenov: Continuous Selections of Multivalued Mappings, Springer 1998)

$\endgroup$
1
  • 1
    $\begingroup$ Just to add more precise reference, this is Theorem 1.2 in Part B of the book. $\endgroup$ Commented Jan 15, 2017 at 19:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.