Paracompactness and Open Relations

TL;DR

This paper explores the relationship between paracompactness, open relations, and selection principles in topology, providing new characterizations and simpler proofs for classical results involving normal and paracompact spaces.

Contribution

It introduces a novel characterization of ca4a0-paracompact normal spaces using selections for convex-valued open relations, simplifying existing proofs.

Findings

Characterization of ca4a0-paracompact normal spaces via open relations

Equivalent conditions for insertions and selections in open relations

Simplified proofs of classical theorems in topology

Abstract

The countably paracompact normal spaces were characterised by Dowker and Kat\v{e}tov in terms of an insertion property. Dowker also characterised them by normality of their product with the closed unit interval. Michael used the Dowker-Kat\v{e}tov insertion property to motivate his selection characterisation of these spaces. Morita extended in a natural way Dowker's product characterisation to all $\tau$-paracompact normal spaces. In this paper, we look at these results from the point of view of open relations. Insertions and selections are equivalent for such relations. Furthermore, we obtain a natural characterisation of $\tau$-paracompact normal spaces in terms of selections for convex-valued open relations. Based on this characterisation, we give simple alternative proofs of the above mentioned results. Other applications are obtained as well.