Between countably compact and ω-bounded

Given a property P of subspaces of a T-1 space X, we say that X is P-bounded if every subspace of X with property P has compact closure in X. Here we study P-bounded spaces for the properties P is an element of {omega D, omega N,C-2} where omega D equivalent to "countable discrete", omega N "countable nowhere dense", and C-2 equivalent to "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: Regular omega D-bounded spaces of Lindelof degree < cov(M) are omega-bounded. Regular omega D-bounded spaces of countable tightness are omega-bounded, and if b > omega(1) then even omega-bounded. If a product of Hausdorff spaces is omega D-bounded then all but one of its factors must be omega-bounded. Any product of at most t many omega D-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated. (C) 2015 Elsevier B.V. All rights reserved.