On \(\delta\)-continuous selections of small multifunctions and covering properties. (English) Zbl 0734.54010

Fedeli, Alessandro; Pelant, Jan

Commentat. Math. Univ. Carol. 32, No. 1, 155-159 (1991).

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.

Reviewer: H.Schirmer (Ottawa)

MSC:

54C60	Set-valued maps in general topology
54C65	Selections in general topology
54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords:

\(\delta\)-continuous selections; small multifunctions; paracompactness; orthocompactness