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On boundary avoiding selections and some extension theorems. (English) Zbl 1059.54020

It is proved that the following conditions are equivalent:
(i) \(X\) is normal and countably paracompact;
(ii) For every convex subset \(C\) of a separable Banach space \(B\), every lsc multifunction \(F: X\to C\) with values convex and compact in \(B\) and every \(F_\sigma\)-subset of \(X\) contains in \(F^-(\text{Int\,}C)\) there exists a continuous selection \(f\) for \(F\) with \(A\subset f^{-1}(\text{Int\,}C)\subset F^-(\text{Int\,}C)\);
(iii) For every \(F: X\to C\) with closed and convex values in \(B\) the thesis of (ii) holds.
If \(X\) is paracompact space, then a similar selection theorem without separability assumptions imposed on \(B\) is true.
As a corollary some results concerning extensions of products and of disjoint families of single-valued functions are obtained. A counterexample giving a solution to a question raised in [M. Frantz, Pac. J. Math. 169, No. 1, 53–73 (1995; Zbl 0843.54024)] is also constructed.

MSC:

54C65 Selections in general topology
54D99 Fairly general properties of topological spaces
26E25 Set-valued functions
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)

Citations:

Zbl 0843.54024
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