Intrinsic characterizations of $C$-realcompact spaces

TL;DR

This paper characterizes $c$-realcompact spaces using $c$-stable families of closed sets and shows how any space can be densely extended to a $c$-realcompact space, introducing related $CP$-compact spaces.

Contribution

It provides a new characterization of $c$-realcompact spaces via $c$-stable families and explores their extensions and related $CP$-compact spaces.

Findings

Characterization of $c$-realcompact spaces using $c$-stable families with finite intersection property.

Any topological space can be densely extended to a $c$-realcompact space.

Introduction of $CP$-compact spaces and their relation to $c$-realcompact spaces.

Abstract

$c$-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41(8), 2018, 1135-1167. We offer a characterization of these spaces $X$ via $c$-stable family of closed sets in $X$ by showing that $X$ is $c$-realcompact if and only if each $c$-stable family of closed sets in $X$ with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a $c$-realcompact space. We show that each topological space can be extended as a dense subspace to a $c$-realcompact space with some desired extension property. An allied class of spaces viz $CP$-compact spaces akin to that of $c$-realcompact spaces are introduced. The paper ends after examining how far a known class of $c$-realcompact spaces could be realized as $CP$-compact for appropriately chosen ideal $P$ of closed sets in $X$.