Selections, Extensions and Collectionwise Normality

TL;DR

This paper explores the relationship between Michael's selection theorem and Dowker's extension theorem in collectionwise normal spaces, providing new proofs and applications for set-valued mappings.

Contribution

It shows that Michael's selection theorem can be reduced to Dowker's extension theorem and offers a simplified proof along with additional applications.

Findings

Reduction of Michael's theorem to Dowker's extension theorem

A new, simpler proof of Michael's selection theorem

Potential applications in topology and set-valued analysis

Abstract

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.