P-paracompact and P-metrizable spaces

摘要

Let P be a directed set and X a space. A collection C of subsets of X is P-locally finite if C = boolean OR{C-p : p is an element of P} where (i) if p <= p' then C-p <= C-p, and (ii) each C-p 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 x 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), the set of all compact subsets of a separable metrizable space M ordered by set inclusion. (C) 2015 Elsevier B.V. All rights reserved.