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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

##### MSC:
 54C65 Selections in general topology 54C60 Set-valued maps in general topology
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##### References:
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