# 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 $$mathbb R^m$$.