$P$-spaces and intermediate rings of continuous functions

Abstract

A completely regular topological space $X$ is called a $P$-space if every zero-set in $X$ is open. An intermediate ring is a ring $A(X)$ of real-valued continuous functions on $X$ containing all the bounded continuous functions. In this paper, we find new characterizations of $P$-spaces $X$ in terms of properties of correspondences between ideals in $A(X)$ and $z$-filters on $X$. We also show that some characterizations of $P$-spaces that are described in terms of properties of $C(X)$ actually characterize $C(X)$ among intermediate rings on $X$.