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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.

MSC:

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

Citations:

Zbl 0973.54021
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References:

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