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Selections that characterize topological completeness. (English) Zbl 0861.54016
For the following properties of a metrizable space \(Y\), it is shown that the previously known implications (a)\(\to\)(b) and (a)\(\to\)(c) are actually equivalences. It is also shown that (d)\(\to\)(a). (Note: \({\mathcal F}(Y)= \{E\subset Y: E\neq\emptyset\), \(E\) closed}). (a) \(Y\) is completely metrizable. (b) If \(X\) is metrizable with \(\dim X=0\), every l.s.c. \(\varphi: X\to {\mathcal F} (Y)\) has a continuous selection. (c) If \(X\) is metrizable and \(\varphi: X\to {\mathcal F} (Y)\) is l.s.c., then there is an u.s.c. \(\psi: X\to {\mathcal F} (Y)\) with \(\psi(x) \subset \varphi(x)\) and \(\psi(x)\) compact for all \(x\in X\). (d) If \(X\) is \({\mathcal F} (Y)\) with the Vietoris topology, then the identity map \(\varphi: X\to {\mathcal F} (Y)\) has a continuous selection. There are also other interesting results.

MSC:
54C65 Selections in general topology
54E50 Complete metric spaces
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