  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.
