On boundary avoiding selections and some extension theorems.
type: Item
instance of: scholarly article
title: On boundary avoiding selections and some extension theorems.
zbMATH Open document ID: 1059.54020
author: Stoyu Barov
author: Jan J. Dijkstra
published in: Pacific Journal of Mathematics
publication date: 2002-10-22T00:00:00Z
review text: 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 [\textit{M. Frantz}, Pac. J. Math. 169, No. 1, 53--73 (1995; Zbl 0843.54024)] is also constructed.
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 26E25
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 1818960
zbMATH Keywords: continuous selection
zbMATH Keywords: continuous extension
zbMATH Keywords: lsc multifunction
zbMATH Keywords: product functions
reviewed by: Włodzimierz Ślęzak
MaRDI profile type: Publication
full work available at URL: pjm.2002.203.79
OpenAlex ID: W1997418362
DOI: 10.2140/PJM.2002.203.79
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description: scientific article; zbMATH DE number 1818960