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

Zbl 0973.54021
Full Text:

References:

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