About extension of upper semicontinuous multi-valued maps and applications

TL;DR

This paper extends the Tietze-Urysohn theorem to multi-valued upper semicontinuous maps with convex values, enabling broader applications in topology, fixed point theory, and game theory.

Contribution

It introduces a multi-valued extension theorem for upper semicontinuous maps with convex values on normal spaces, generalizing classical extension results.

Findings

Extension of multi-valued maps preserves upper semicontinuity and convexity.

Applications to extending semicontinuous functions and characterizing normal spaces.

Establishment of fixed point theorems and equilibrium existence in game theory.

Abstract

We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Precisely, we prove that any upper semicontinuous multi-valued map with nonempty closed convex values defined on a closed subset (resp. closed perfectly normal subset) of a completely normal (resp. of a normal) space $X$ into the unit interval $[0,1]$ can be extended to the whole space $X$. The extension is upper semicontinuous with nonempty closed convex values. We apply this result for the extension of real semicontinuous functions, the characterization of completely normal spaces, the existence of Gale-Mas-Colell and Shafer-Sonnenschein type fixed point theorems and the existence of equilibrium for qualitative games.