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Michael selection theorem under weak lower semicontinuity assumption. (English) Zbl 0706.54018
Suppose that X is paracompact, Y a Banach space, and \(\phi: X\to 2^ Y\) with each \(\phi\) (x) closed and convex. The reviewer proved in Ann. Math. 63, 361-382 (1956; Zbl 0071.159) that, if \(\phi\) is lower semicontinuous, then \(\phi\) has a continuous selection. (In fact, \(\phi\) is lower semicontinuous if and only if, for every closed (or every singleton) \(A\subset X\), every continuous selection for \(\phi| A\) extends to a continuous selection for \(\phi\).) The paper under review obtains a new sufficient condition (called weakly lower semicontinuous) for \(\phi\) to have a continuous selection which simultaneously weakens both lower semicontinuous and another sufficient condition (called weakly Hausdorff lower semicontinuous) obtained by F. S. De Blasi and J. Myjak in Proc. Am. Math. Soc. 93, 369-372 (1985; Zbl 0565.54013).
Reviewer: E.Michael

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