Topological Properties of the Space of Convex Minimal Usco Maps

Abstract

Let X be a Tychonoff space and M C(X) be the space of convex minimal usco maps with values in R, the space of real numbers. Such setvalued maps are important in the study of subdifferentials of convex functions. Using the strong Choquet game we prove complete metrizability of M C(X) with the upper Vietoris topology. If X is normal, elements of M C(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 M C(X).