Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space
The main object of this paper are topologies on the collection of \(\text{CL}(X)\) consisting of all closed nonempty subsets of an ultrametric space \(X\). V. Gutev has proved that, for a separable complete ultrametric space \(X\), there exists a continuous selection from \(\text{CL}(X)\) to \(X\) if \(\text{CL}(X)\) is endowed with the so-called Ball-topology. In other words, there exists a continuous map \(f\) from \(\text{CL}(X)\) to \(X\) such that \(f(C)\in C\), for each \(C\in\text{CL}(X)\). Here, the authors characterize those separable complete ultrametric spaces for which Gutev's result extends to the much more common and useful Wijsman-topology \(W_d\) on \(\text{CL}(X)\). One of the main results of the paper states that, for a separable complete ultrametric space \(X\), \(\text{CL}(X)\) admits a continuous selection iff \((\text{CL}(X),W_d)\) is totally disconnected. In my opinion, the methods and examples developed in this paper will be of interest for many other questions related to hyperspaces of non-archimedean topological spaces in general, too. (Compare authors like P. Nyikos, the reviewed ones, and others).
scientific article; zbMATH DE number 1307939