Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space
type: Item
instance of: scholarly article
title: Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space
zbMATH Open document ID: 0944.54009
DOI: 10.1016/S0166-8641(97)00175-2
author: Daniela Bertacchi
author: Camillo Costantini
published in: Topology and its Applications
publication date: 1999-09-06T00:00:00Z
review text: 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).
reviewed by: Hans-Christian Reichel
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 1307939
zbMATH Keywords: Wijsman-topology
MaRDI profile type: Publication
cites work: Q3920292
cites work: Q4277838
cites work: Topological well-ordering and continuous selections
cites work: Q3994935
cites work: Selected Selection Theorems
cites work: Selections and hyperspace topologies via special metrics
cites work: Equivalence among hypertopologies
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description: scientific article; zbMATH DE number 1307939