Topological spaces compact with respect to a set of filters

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.