A reverse viewpoint on the upper semicontinuity of multivalued maps

Abstract

For a multivalued map $\varphi \colon Y\multimap (X,\tau )$ between topological spaces, the upper semifinite topology $\mathcal {A}(\tau )$ on the power set $\mathcal {A}(X)=\{A\subset X \colon A\neq \emptyset \}$ is such that $\varphi $ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi \colon Y\rightarrow (\mathcal {A}(X),\mathcal {A}(\tau ))$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma $ on $\mathcal {A}(X)$, we consider a natural topology $\mathcal {R}(\Gamma )$ on $X$, constructed from $\Gamma $ satisfying $\mathcal {R}(\Gamma )=\tau $ if $\Gamma =\mathcal {A}(\tau )$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map $\varphi \colon Y\multimap (X,\mathcal {R}(\Gamma ))$ to be equivalent to the continuity of the singlevalued map $\varphi \colon Y\rightarrow (\mathcal {A}(X),\Gamma )$.