Continuous Selections, Function Spaces and Partitions of Unity

TL;DR

This paper explores the relationships between partitions of unity, continuous selections, and paracompactness, providing equivalent formulations of Michael's selection theorem without explicitly referencing paracompactness.

Contribution

It presents new equivalent forms of Michael's selection theorem, linking partitions of unity to continuous selections in function spaces without explicitly mentioning paracompactness.

Findings

Partitions of unity are equivalent to continuous selections in certain function spaces.

The existence of partitions of unity characterizes paracompactness.

Equivalent formulations of the Michael selection theorem are established.

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.