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:
- 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.
- 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.
