Between countably compact and ω-bounded

Given a property P   of subspaces of a T1T1 space X, we say that X is 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∈{ωD,ωN,C2}P∈{ωD,ωN,C2} where ωD ≡ “countable discrete”, ωN   ≡ “countable nowhere dense”, and C2C2 ≡ “second countable”. Clearly, for each of these P-bounded is between countably compact and ω-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 ωD-bounded spaces which show that ωD-boundedness is much stronger than countable compactness:
• Regular ωD  -bounded spaces of Lindelöf degree • Regular ωD-bounded spaces of countable tightness are ωN  -bounded, and if b>ω1b>ω1 then even ω-bounded.
• If a product of Hausdorff spaces is ωD-bounded then all but one of its factors must be ω-bounded.
• Any product of at most tt many ωD-bounded spaces is countably compact.As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.