Selections and hyperspace topologies via special metrics (Q1916471)

Starting from the well-known result of \textit{R. Engelking}, \textit{R. W. Heath} and \textit{E. Michael} [Invent. Math. 6, 150-158 (1968; Zbl 0167.20504), see also Lect. Notes Math. 171, 8-11 (1970; Zbl 0207.21202)] which says that any zero-dimensional completely metrizable space \(Y\) admits a continuous selection with respect to the Vietoris topology on the collection \({\mathcal F}(Y)\) of all nonempty closed subsets of \(Y\), the author proves that a complete metric space \(Y\) with a non-Archimedean metric \(d\) admits a continuous selection \(f:{\mathcal F}(Y)\to Y\) with respect to the \(d\)-proximal topology, which is continuous also within both Hausdorff and Vietoris topologies on \({\mathcal F}(Y)\).