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 $\bmod \;k$. A space X has a refining sequence $\bmod \;k$ if there exists a sequence $\{ {\mathcal {G}_n}|n \in N\}$ of open covers for X such that $\cap _{n = 1}^\infty {\text {St}}(C,{\mathcal {G}_n}) = P_C^1$ is compact for each compact subset C of X and ${\text {\{ St}}{(C,{\mathcal {G}_n})^ - }|n \in N\}$ is a neighborhood base for $P_C^1$. If $P_C^1 = C$ 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 $P_C^1 = C$ for all such sets then X is developable. Thus the concept of a refining sequence $\bmod \;k$ is natural and it is helpful in understanding paracompact p-spaces.