reference request – Does a generalization of Tietze’s extension theorem hold for set-valued functions?


Let $X$ be a normal topological space. Tietze’s extension theorem says that if $A subset X$ is closed, then a continuous function $f: A to mathbb R^n$ can be extended to a continuous function whose domain is all of $X$. The theorem generalizes to functions taking values in any locally convex linear space (see this).

I am wondering if the theorem holds for certain set-valued functions.

In particular, let $phi$ be a function from closed $A subset X$ into compact subsets of $mathbb R^n$. To say that $phi$ is continuous on $A$ means that the following conditions are met for every $x in A$:

(1) For every neighborhood $U$ of $phi(x)$, there is a neighborhood $V$ of $x$ such that $phi(y) subset U$ for all $y in V$;

(2) For every open subset $U$ of $A$ for which $phi(x) cap U neq emptyset$, there is a neighborhood of $V$ of $x$ such that $phi(y) cap U neq emptyset$ for all $y in V$.

Can $phi$ be extended to a continuous set-valued function whose domain is all of $X$?

In principle, I don’t mind assuming that $X$ is actually a subset of $mathbb R^m$.