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

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.