Finite-valued multiselections (Q529086)

Let \(\mathcal{F}(X)\) be the collection of all nonempty closed subsets of a space \(X\). Each subcollection \(\mathcal{D}\subset\mathcal{F}(X)\) will carry the Vietoris topology \(\tau_{V}\). A space \(X\) is weakly orderable if it has a coarser open interval topology. In the present paper, the author is interested in set-valued selections for hyperspaces. The main results are: Theorem 1.1. Let \(X\) be a connected space which has a \(\tau_{V}\)-continuous multiselection \(\varphi:\mathcal{F}_{n}(X)\rightarrow\mathcal{F}_{n+1}(X)\) for some \(n\geq 1\). Then \(X\) is weakly orderable. Theorem 1.2. Let \(X\) be a connected space which has a \(\tau_{V}\)-continuous multiselection \(\phi:\mathcal{F}(X)\rightarrow\sum(X)\). If \(\varphi\) is not singleton-valued at some element of \(\mathcal{F}(X)\), then \(X\) is compact and orderable.