Unified Selection Theorems

Abstract

Let X be a paracompact space and let Z be one of its closed, zero-dimensional subsets; hence dim Z = 0. Then each lower semicontinuous mapping F from X into a Banach space B with closed convex values F(x), for all x ∉ Z, admits a singlevalued selection. To see this, it suffices to use Zero-dimensional selection theorem (A.2.4) for the restriction F∣ Z , and then use Convex-valued theorem (A.1.5) for a lower semicontinuous mapping, which coincides with F over X\Z and which coincides with the chosen singlevalued selection of F∣ Z onto Z.