Topological Properties of the Space of Convex Minimal Usco Maps

TL;DR

This paper investigates the topological structure of the space of convex minimal usco maps, proving its complete metrizability and approximation properties, which are relevant for convex analysis and subdifferential studies.

Contribution

It establishes the complete metrizability of the space of convex minimal usco maps and explores approximation by continuous functions under certain conditions.

Findings

MC(X) is completely metrizable with the upper Vietoris topology

Elements of MC(X) can be approximated by continuous functions if X is normal

The paper analyzes countability properties of the upper Vietoris topology on MC(X)

Abstract

Let X be a Tychonoff space and MC(X) be the space of convex minimal usco maps with values in R, the space of real numbers. Such set-valued maps are important in the study of subdifferentials of convex functions. Using the strong Choquet game we prove complete metrizability of MC(X) with the upper Vietoris topology. If X is normal, elements of MC(X) can be approximated in the Vietoris topology by continuous functions. We also study first countability, second countability and other properties of the upper Vietoris topology on MC(X).