Selections and hyperspaces
Given a Hausdorff topological space \(X\), let \(\mathcal F(X)\) be the set of all nonempty closed subsets of \(X\), and denote by \(\tau_V\) the \textit{Vietoris topology} on \(\mathcal F(X)\) having as a base sets of the form \(\{S\in \mathcal F(X): S\subset \bigcup \mathcal V \;\text{and} \;S\cap V\neq \emptyset \;\text{whenever} \;V\in\mathcal V\}\), where \(\mathcal V\) runs through the finite families of nonempty open subsets of \(X\). For \(\mathcal D\subseteq \mathcal F(X)\), the map \(f:\mathcal D\to X\) is a \textit{selection} for \(\mathcal D\) provided \(f(S)\in S\) for each \(S\in \mathcal D\), and \(f\) is a \textit{\(\tau_V\)-continuous selection} for \(\mathcal D\), provided \(f\) is continuous with respect to the relative Vietoris topology on \(\mathcal D\); more generally, one can define the notion of \textit{\(\tau\)-continuous selections}, if \(\tau_V\) is replaced by other hyperspace topologies \(\tau\) on \(\mathcal D\).NEWLINENEWLINE In this survey paper the author reviews the status of the so-called \textit{hyperspace selection problem}: when is it possible to find a \(\tau\)-continuous selection for \(\mathcal D\), where \(\tau\) stands for various hypertopologies (such as the Vietoris, Fell, or Wijsman, respectively), and \(\mathcal D\) is \(\mathcal F(X)\), the nonempty compact subsets of \(X\) or, for a given \(n\), the up to \(n\)-element subsets of \(X\), respectively. The notable solutions are surveyed according to the properties of the spaces studied, namely, for connected-like, compact-like, disconnected-like, and metrizable spaces, respectively.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].
scientific article; zbMATH DE number 6329563