  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.
