The Set-Self-Tietze Property

TL;DR

This paper introduces the set-self-Tietze property, extending the classical self-Tietze property to upper semi-continuous set-valued functions, and proves that all compact metric spaces have this property while the torus does not.

Contribution

It defines the set-self-Tietze property for topological spaces and establishes that all compact metric spaces possess this property, unlike the torus.

Findings

All compact metric spaces are set-self-Tietze.

The torus does not have the 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.