Paracompactness and Open Relations

Abstract

The countably paracompact normal spaces were characterised by Dowker and Katětov in terms of an insertion property. Dowker also characterised them by normality of their product with the closed unit interval. Michael used the Dowker-Katětov insertion property to motivate his selection characterisation of these spaces. Morita extended in a natural way Dowker's product characterisation to all $τ$-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 $τ$-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.

Theorems & Definitions (32)

• Theorem 1.1
• Proposition 2.1
• proof
• Theorem 2.2
• proof : Proof of Theorem \ref{['theorem-Count-Par-v2:1']}
• Corollary 2.3
• proof
• Proposition 3.1
• proof
• Proposition 3.2