P-paracompact and P-metrizable spaces

Let P be a directed set and X   a space. A collection CC of subsets of X is P-locally finite   if C=⋃{Cp:p∈P}C=⋃{Cp:p∈P} where (i) if p≤p′p≤p′ then Cp⊆Cp′Cp⊆Cp′ and (ii) each CpCp is locally finite. Then X is P-paracompact if every open cover has a P-locally finite open refinement. Further, X is P-metrizable   if it has a (P×N)(P×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 K(M)K(M), the set of all compact subsets of a separable metrizable space M ordered by set inclusion.