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Continuous selections for weakly Hausdorff lower semicontinuous multifunctions. (English) Zbl 0565.54013
Let us denote the set of all nonempty closed convex sets of the Banach space Y by \(C_ c(Y)\), and let us say that the multifunction F from a topological space X to \(C_ c(Y)\) is weakly Hausdorff lower semicontinuous \((H_ w-l.s.c.)\) if for all \(x_ 0\in X\), for all \(\epsilon >0\), and for all V neighbourhood of \(x_ 0\) there exists a neighbourhood U of \(x_ 0\), \(U\subset V\), and there exists x’\(\in U\) such that for all \(x\in U\) we obtain \(F(x')\subset F(x)+\epsilon S\), where S is the unit sphere in Y. A geometrical lemma enables us to use a result of Deutsch and Kenderov to deduce the following result: Let X be paracompact. Then every \(H_ w-l.s.c\). multifunction \(F: X\to C_ c(Y)\) admits a continuous selection. This theorem is similar to Michael’s selection theorem but does not follow from it.
Reviewer: P.Holicky

54C65 Selections in general topology
54C60 Set-valued maps in general topology
Full Text: DOI
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