van Mill, Jan; Pelant, Jan; Pol, Roman Selections that characterize topological completeness. (English) Zbl 0861.54016 Fundam. Math. 149, No. 2, 127-141 (1996). 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. Reviewer: E.Michael (Seattle) Cited in 19 Documents MSC: 54C65 Selections in general topology 54E50 Complete metric spaces Keywords:completely metrizable space; Vietoris topology PDFBibTeX XMLCite \textit{J. van Mill} et al., Fundam. Math. 149, No. 2, 127--141 (1996; Zbl 0861.54016) Full Text: EuDML