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 $X$.

It is also true on the space of all closed subsets if $X,Y$ are bounded metric spaces and $F$ is uniformly continuous, given the Hausdorff topology. However I have been unable to track down any clear references as to what happens in the more general case where we are concerned with all closed subsets and do not know about the uniform continuity of $F.

Is there a topology on $2^X$ (agreeing with the Hausdorff topology when $X$ is compact) for continuous maps between spaces induce continuous maps between their associated hyperspaces? Any examples or references would be greatly appreciated!

  • $\begingroup$ I learned from Steve Vickers, Chris Townsend and Alexander Kurz that for general spaces the "correct" Vietoris is the set of convex subsets ( = intersections of a closed and a compact saturated subset (saturated = intersection of opens)) $\endgroup$ – მამუკა ჯიბლაძე Oct 14 '17 at 6:22

The following modification of the Vietoris topology seems to satisfy the requirements.

Let $\tau$ be the topology on $2^X$ consisting of the sets $\mathcal U\subset 2^X$ such that for every closed set $F\in\mathcal U$ there are open sets $U_1,\dots,U_n\subset X$ intersecting $F$ and a continuous pseudometric $d$ on $X$ such that the set $$\bigcap_{i=1}^n\{E\in 2^X:E\cap U_i\ne\emptyset\}\cap\{E\in 2^X:E\subset \bigcup_{x\in F}B_d(x,1)\}$$ is contained in $\mathcal U$.

Here $B_d(x,1)=\{y\in X:d(x,y)<1\}$ is the open 1-ball cenetered at $x$.


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