A generalization of the notion of a \(P\)-space to proximity spaces (Q2850662)

In this note the author introduces the notion of \(P_{\aleph_{1}}\)-proximity, so generalizing the notion of a \(P\)-space to proximity spaces. A Tychonoff space \(X\) is a \(P\)-space when every zero-set is open. A \(P_{\aleph_{1}}\)-proximity space \((X, \delta)\) is defined as an Efremovič proximity space where the proximity \(\delta\) satisfies the further following property: if \(A_{n} \prec B \) for all \(n \in \mathbb{N}\), then \(\cup A_{n} \prec B,\) where \(\prec\) is the natural strong inclusion associated with \(\delta.\) The author connects \(P_{\aleph_{1}}\)-proximity spaces to reduced algebras of sets, by showing that the category of zero-dimensional proximity spaces (i.e. a proximity space \((X,\delta)\) is zero-dimensional when if \(A \prec B,\) then there exists \(C\) so that \(A \prec C \prec C \prec B \)) with proximity maps is isomorphic to the category of reduced algebras of sets with measurable maps and their correspondent subcategories of \(P_{\aleph_{1}}\)-proximity spaces and reduced algebras \(\sigma\)-algebras are isomorphic as well. Moreover, the author gives a functional characterization of the \(P_{\aleph_{1}}\)-proximity spaces as those proximity spaces for which the pointwise limit of a sequence of proximity functions to \([0,1]\) is also a proximity function; so obtaining, as a corollary, that if \(X\) is Tychonoff and \(\delta\) is the proximity induced by the Stone-Čech compactification of \(X\), then \(X\) is a \(P\)-space if and only if \((X,\delta)\) is a \(P_{\aleph_{1}}\)-proximity space. The paper ends with the demonstration that it is sometimes better to consider \(\sigma\)-algebras as \(P_{\aleph_{1}}\)-proximities, since proximity spaces appear easier to work with than \(\sigma\)-algebras.