Berge's Theorem for Noncompact Image Sets

TL;DR

This paper extends Berge's theorem, which guarantees lower semi-continuity of minima over image sets, to cases where the image sets are not necessarily compact, broadening its applicability in topological analysis.

Contribution

It generalizes Berge's theorem to noncompact image sets in Hausdorff spaces, providing new insights into the properties of minima in such contexts.

Findings

Extended Berge's theorem to noncompact image sets

Analyzed properties of minima in noncompact settings

Broadened applicability of semi-continuity results

Abstract

For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge's theorem states lower semi-continuity of the minimum of this function taken over the image sets. It assumes that the image sets are compact. For Hausdorff topological spaces, this paper extends Berge's theorem to set-valued mappings with possible noncompact image sets and studies relevant properties of minima.