# Induced maps on Hyperspace Topologies

If $X$ is a topological space let $2^X$ denote the set of closed subsets. There are multiple topologies one may equip $2^X$ with (in particular, I have in mind the Vietoris, Fell and similar topologies which agree with the topology induced by the Hausdorff metric when $X$ is compact), and I have been trying to track down references to answer the following question: if $F:X\to Y$ is continuous then for which topologies on $2^X,2^Y$ is the induced map $F_\ast:2^X \to 2^Y$ given by $Z\mapsto \overline{F(Z)}$ continuous?

If we are interested only in putting a topology on the space of compact subsets of $X,Y$ (denoted $K_X,K_Y$) then the induced map $F_\ast:K_X\to K_Y$ given by $Z\mapsto F(Z)$ is continuous with respect to the Vietoris topology, having a subbasis of sets of the form

$$\mathcal{O}_C=\{Z\in K_X\;\mid\; Z\cap C=\varnothing\}\hspace{1cm} \mathcal{O}'_U=\{Z\in K_X\;\mid\; Z\cap U\neq\varnothing\}$$ for $C$ closed and $U$ open in $X$.