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.