Between countably compact and $\omega$-bounded

Abstract

Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{\omega D, \omega N, C_2 \}$ where $\omega D \, \equiv$ "countable discrete", $\omega N \, \equiv$ "countable nowhere dense", and $C_2 \,\equiv$ "second countable". Clearly, for each of these $P$-bounded is between countably compact and $\omega$-bounded. We give examples in ZFC that separate all these boundedness properties and their appropriate combinations. Consistent separating examples with better properties (such as: smaller cardinality or weight, local compactness, first countability) are also produced. We have interesting results concerning $\omega D$-bounded spaces which show that $\omega D$-boundedness is much stronger than countable compactness: $\bullet$ Regular $\omega D$-bounded spaces of Lindel\"of degree $< cov(\mathcal{M})$ are $\omega$-bounded. $\bullet$ Regular $\omega D$-bounded spaces of countable tightness are $\omega N$-bounded, and if $\mathfrak{b} > \omega_1$ then even $\omega$-bounded. $\bullet$ If a product of Hausdorff space is $\omega D$-bounded then all but one of its factors must be $\omega$-bounded. $\bullet$ Any product of at most $\mathfrak{t}$ many Hausdorff $\omega D$-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.