Barov, Stoyu; Dijkstra, Jan J. On boundary avoiding selections and some extension theorems. (English) Zbl 1059.54020 Pac. J. Math. 203, No. 1, 79-87 (2002). 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. Reviewer: Wlodzimierz Ślȩzak (Bydgoszcz) Cited in 2 ReviewsCited in 3 Documents MSC: 54C65 Selections in general topology 54D99 Fairly general properties of topological spaces 26E25 Set-valued functions 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) Keywords:continuous selection; continuous extension; lsc multifunction; product functions Citations:Zbl 0843.54024 PDFBibTeX XMLCite \textit{S. Barov} and \textit{J. J. Dijkstra}, Pac. J. Math. 203, No. 1, 79--87 (2002; Zbl 1059.54020) Full Text: DOI