An internal characterization of paracompact 𝑝-spaces

Abstract

The purpose of this paper is to characterize paracompact p-spaces in terms of spaces with refining sequences mod k \bmod \;k . A space X has a refining sequence mod k \bmod \;k if there exists a sequence { G n | n ∈ N } \{ {\mathcal {G}_n}|n \in N\} of open covers for X such that ∩ n = 1 ∞ St ( C , G n ) = P C 1 \cap _{n = 1}^\infty {\text {St}}(C,{\mathcal {G}_n}) = P_C^1 is compact for each compact subset C of X and { St ( C , G n ) − | n ∈ N } {\text {\{ St}}{(C,{\mathcal {G}_n})^ - }|n \in N\} is a neighborhood base for P C 1 P_C^1 . If P C 1 = C P_C^1 = C for each compact subset C of X then X is metrizable. On the other hand if we restrict the set C to the family of finite subsets of X in the above definition then we have a characterization for strict p-spaces. Moreover, in this case, if P C 1 = C P_C^1 = C for all such sets then X is developable. Thus the concept of a refining sequence mod k \bmod \;k is natural and it is helpful in understanding paracompact p-spaces.