Approaching points by continuous selections. (English) Zbl 1111.54020

In this paper the author continues investigations from the paper [V. Gutev and T. Nogura, Proc. Am. Math. Soc. 129, 2809–2815 (2001; Zbl 0973.54021)].
Let \(X\) be a topological space. Let \({\mathcal F}(X)\) be the hyperspace of all nonempty closed subsets of \(X\) equipped with the Vietoris topology. A map \(f: {\mathcal F}(X) \rightarrow X\) is a selection for \({\mathcal F}(X)\) if \(f(S) \in S\) for every \(S \in {\mathcal F}(X)\). The purpose of this paper is to establish some further results about spaces \(X\) for which the set \(\{f(X): f \) is a continuous selection for \({\mathcal F}(X)\}\) is dense in \(X\). It is proved that this set is dense if and only if \(X\) has a clopen \(\pi\)-base. Several applications follow by this characterization. For example, it is demonstrated that a homogeneous separable metrizable space \(X\) has a continuous selection for its hyperspace \({\mathcal F}(X)\) if and only if one of the following holds:
a) \(X\) is a discrete space,
b) \(X\) is a discrete sum of copies of the Cantor set,
c) \(X\) is the irrational line.


54C65 Selections in general topology
54B20 Hyperspaces in general topology
54F65 Topological characterizations of particular spaces


Zbl 0973.54021
Full Text: DOI Euclid


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