Topological spaces compact with respect to a set of filters

TL;DR

This paper introduces a generalized notion of compactness in topological spaces based on families of filters, unifying various compactness concepts and analyzing their product stability and the role of ultrafilters.

Contribution

It characterizes when product stability of sequencewise -compactness holds, linking it to the existence of an ultrafilter within the family of filters.

Findings

Product stability of sequencewise -compactness occurs iff it reduces to an F-compactness for some filter F.

If stable and with a non-singleton space, F must be an ultrafilter.

Detailed analysis of sequential and --compactness cases.

Abstract

If $\mathcal P$ is a family of filters over some set $I$, a topological space $X$ is \emph{sequencewise $\mathcal P$-\brfrt compact} if, for every $I$-indexed sequence of elements of $X$, there is $F \in \mathcal P$ such that the sequence has an $F$-limit point. Countable compactness, sequential compactness, initial $\kappa$-compactness, $[ \lambda ,\mu]$-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise $\mathcal P$-compactness, for appropriate choices of $\mathcal P$. We show that sequencewise $\mathcal P$-compactness is preserved under taking products if and only if there is a filter $F \in \mathcal P$ such that sequencewise $\mathcal P$-compactness is equivalent to $F$-compactness. If this is the case, and there exists a sequencewise $\mathcal P$-compact $T_1$ topological space with more than one point, then $F$ is necessarily an ultrafilter. The particular cases of sequential compactness and of $[ \lambda ,\mu]$-compactness are analyzed in detail.