Continuous Selections, Function Spaces and Partitions of Unity

Abstract

The famous Michael selection theorem deals with the characterisation of paracompact spaces by continuous selections of lower semi-continuous mappings in Banach spaces. In this paper, we will discuss several equivalent forms of this theorem, without explicitly mentioning paracompactness. This will be based on a previous result, also obtained by Michael, that a space $X$ is paracompact if and only if every open cover of $X$ has an index-subordinated partition of unity. Thus, we will show that the existence of such partitions of unity on a space $X$ is equivalent to the existence of continuous selections for special lower semi-continuous mappings from $X$ to the nonempty convex subsets of special function spaces.