The Set-Self-Tietze Property

Abstract

We introduce the set-self-Tietze property, an analogue of the self-Tietze property for upper semi-continuous set-valued functions. A topological space $X$ is self-Tietze, if for every closed $A \subseteq X$ and continuous function $f \colon A \to X$, there is a continuous extension $F \colon X \to X$ of $f$. A topological space $X$ is set-self-Tietze, if for every closed $A \subseteq X$ and upper semi-continuous set-valued function $f \colon A \to 2^X$, there exists an upper semi-continuous set-valued function $F \colon X \to 2^X$ such that $\left. F \right|_A = f$. We show every compact metric space is set-self-Tietze, and that the torus is not self-Tietze.