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

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.

💡 Research Summary

The paper extends the classical notion of a P‑space—originally defined in topology as a space where countable unions of open sets remain open—to the broader framework of proximity spaces. The authors introduce the concept of a (P_{\aleph_{1}})‑proximity: a proximity relation (\prec) on a set (X) satisfies the (P_{\aleph_{1}}) condition if for every sequence of subsets ((A_{n})_{n\in\mathbb{N}}) and any subset (B\subseteq X), the implication \