A Generalization of the notion of a $P$-space to proximity spaces

Abstract

In this note, we shall generalize the notion of a $P$-space to proximity spaces and investigate the basic properties of these proximities. We therefore define a $P_{{\aleph }_1}$-proximity to be a proximity where if $A_n\prec B$ for all $n\in \mathbb{N}$, then ${\bigcup }_n{A}_n\prec B$. It turns out that the class of $P_{{\aleph }_1}$-proximities is equivalent to the class of $\sigma$-algebras. Furthermore, the $P_{{\aleph }_1}$-proximity coreflection of a proximity space is the $\sigma$-algebra of proximally Baire sets.