$P$-Paracompact and $P$-Metrizable Spaces

TL;DR

This paper introduces and studies the concepts of $P$-paracompact and $P$-metrizable spaces, generalizing classical notions by using directed sets, with a focus on the case where $P$ is the set of compact subsets of a separable metrizable space.

Contribution

It provides the first detailed analysis of $P$-paracompact and $P$-metrizable spaces, especially when $P$ is the set of compact subsets of a separable metrizable space.

Findings

Introduces $P$-paracompact and $P$-metrizable spaces.

Establishes properties and characterizations for these spaces.

Analyzes the case when $P = \mathcal{K}(M)$ for a separable metrizable space $M$.

Abstract

Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq \mathcal{C}_{p'}$ and (ii) each $\mathcal{C}_p$ is locally finite. Then $X$ is \emph{$P$-paracompact} if every open cover has a $P$-locally finite open refinement. Further, $X$ is \emph{$P$-metrizable} if it has a $(P \times \mathbb{N})$-locally finite base. This work provides the first detailed study of $P$-paracompact and $P$-metrizable spaces, particularly in the case when $P$ is a $\mathcal{K}(M)$, the set of all compact subsets of a separable metrizable space $M$ ordered by set inclusion.