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On $$\delta$$-continuous selections of small multifunctions and covering properties. (English) Zbl 0734.54010
Let Y be a metric space with metric d. Then a function $$f:X\to Y$$ is called $$\delta$$-continuous if there exists, for every $$x\in X$$, an open neighbourhood U(x) so that $$f(U(x))\subset \{y\in Y| \quad d(y,f(x))<\delta \},$$ and a multifunction $$\phi: X\to Y$$ is called $$\delta$$-small if, for every $$x\in X$$, the diameter $$diam(\phi (x))\leq \delta.$$ These two concepts are related. The reviewer has shown [Can. J. Math. 24, 631-635 (1972; Zbl 0219.54012)] that every selection of an lsc or usc $$\delta$$-small multifunction is $$2\delta$$-continuous. She also proved that if X is compact Hausdorff and $$f:X\to Y$$ is $$\delta$$- continuous, then there exist both a point-open lsc and a point-closed usc $$2\delta$$-small multifunction which has f as a selection. The authors of this paper are interested in weakening the compactness assumption of this last result. They show that if X is only paracompact (orthocompact), then there still exists a point-open lsc (point-closed usc) $$2\delta$$-small multifunction which has f as a selection. Here an orthocompact space X is a space for which every open cover $${\mathcal U}$$ has an open refinement $${\mathcal V}$$ such that $$\cap {\mathcal W}$$ is open for any $${\mathcal W}\subset {\mathcal V}$$. the paper contains four informative examples which show that without the assumption of paracompactness (orthocompactness) in some cases point-open lsc (point-closed usc) $$2\delta$$-small multifunctions which have a given $$\delta$$-continuous function as a selection exist, but in other cases they do not. Hence their existence cannot be used to characterize these types of compactness.
##### MSC:
 54C60 Set-valued maps in general topology 54C65 Selections in general topology 54D20 Noncompact covering properties (paracompact, LindelĂ¶f, etc.)
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