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

TL;DR

This paper extends the concept of $P$-spaces to proximity spaces, introduces $P_{ approx_{1}}$-proximities, and explores their properties, revealing their equivalence to $\sigma$-algebras and identifying the coreflection with proximally Baire sets.

Contribution

It generalizes $P$-spaces to proximity spaces, defines $P_{ approx_{1}}$-proximities, and establishes their equivalence to $\sigma$-algebras, linking proximity theory with measure theory.

Findings

$P_{ approx_{1}}$-proximities are equivalent to $\sigma$-algebras.

The coreflection of a proximity space is the $\sigma$-algebra of proximally Baire sets.

The class of $P_{ approx_{1}}$-proximities characterizes certain measure-theoretic structures.

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.