A SELECTION-THEORETIC APPROACH TO CERTAIN EXTENSION THEOREMS

Abstract

This chapter discusses a selection-theoretic approach to certain extension theorems. A function space C(Y, E) can be endowed with several natural topologies: the most familiar are the topology of point-wise convergence, the compact-open topology, and the topology of uniform convergence. Furthermore, it was proved in [HLZ] that even if X is compact, one cannot always obtain extension operators that are continuous in the topology of point-wise convergence. However, it was left open whether one could always obtain extension operators that are continuous with respect to the topology of uniform convergence. Michael's selection theorem [M1, M2] asserts that any lower semicontinuous carrier ψ; Υ → C (B), where Y is paracompact, admits a continuous selection, that is, a continuous function η:Υ → B having the property η(y) ɛ ψ(y) for each y ɛ Y.