Normality-like properties, paraconvexity and selections.

Abstract

In 1956, E. Michael proved his famous convex-valued selection theorems for l.s.c. mappings de ned on spaces with higher separation axioms (paracompact, collectionwise normal, normal and countably paracompact, normal, and perfectly normal), [39]. In 1959, he generalized the convex-valued selection theorem for mappings de ned on paracompact spaces by replacing \convexity" with \ -paraconvexity", for some xed constant 0 < 1 (see, [42]). In 1993, P.V. Semenov generalized this result by replacing with some continuous function f : (0;1) ! [0; 1) (functional paraconvexity) satisfying a certain property called (PS), [63]. In this thesis, we demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well. We also demonstrate that the -paraconvex-valued and the functionally-paraconvex valued selection theorems remain true for C 0 (Y )-valued mappings de ned on -collectionwise normal spaces, where is an in nite cardinal number. Finally, we prove that these theorems remain true for C (Y )-valued mappings de ned on -PF-normal spaces; and we provide a general approach to such selection theorems.