  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.
