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

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 $p\le p^{\prime }$ then ${\mathcal{C}}_p\subseteq {\mathcal{C}}_{p^{\prime }}$ 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.