Relations between minimal usco and minimal cusco maps

Abstract

In our paper we give a characterization of (set-valued) maps which are minimal usco and minimal cusco simultaneously. Let $X$ be a topological space and $Y$ be a Banach space. We show that there is a bijection between the space $\mathrm{MU}(X,Y)$ of minimal usco maps from $X$ to $Y$ and the space $\mathrm{MC}(X,Y)$ of minimal cusco maps from $X$ to $Y$, and we study this bijection with respect to various topologies on underlying spaces. Let $X$ be a Baire space and $Y$ be a Banach space. Then $(\mathrm{MU}(X,Y),{\tau }_U)$ and $(\mathrm{MC}(X,Y),{\tau }_U)$ are homeomorphic, where ${\tau }_U$ is the topology of uniform convergence.