Selections and totally disconnected spaces
All spaces are assumed to be at least Hausdorff, and for a space \(X{\mathcal F}(X)\) denotes the set of all nonempty closed subsets of \(X\). Let \({\mathcal D}\subset{\mathcal F}(X)\). A map \(f:{\mathcal D}\to X\) is called a selection for \({\mathcal D}\) if \(f(S)\in S\) for every \(S\in{\mathcal D}\) and selection \(f:{\mathcal D}\to X\) is said to be continuous if it is continuous with respect to the Vietoris topology on \({\mathcal D}\). The authors demonstrate that for every \(n<\omega\) there exists a separable completely metrizable space \(X_n\) which has a continuous selection for its Vietoris hyperspace of nonempty compact subsets, but \(\dim X_n= n\). The fundamental result in this paper is: Theorem 2.1. Let \(X\) be a totally disconnceted space such that \(l(\Omega(X))\leq \omega\) for some \(n\geq 2\). Then \([X]^n\) has a continuous selection, where \([X]^n= \{S\subset X:|S|= n\}\), \(\Omega(X^n)= \{(x_1, x_2,\dots, x_n)\in X^n: |\{x_1, x_2,\dots, x_n\}|= n\}\) and \(l(\Omega(X^n))\) denotes the Lindelöf number of \(\Omega(X^n)\). In addition, related results and open problems are discussed.
scientific article; zbMATH DE number 5275705