2014/06/30 by Juhász, István, Soukup, Lajos, Szentmiklóssy, Zoltán
#54A25 #54A35 #54D30 #54D65 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.1406.7805
Given a property P of subspaces of a T1 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 ∈ \ωD, ωN, C2 \ where ωD ≡ "countable discrete", ωN ≡ "countable nowhere dense", and C2 ≡ "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: \bullet Regular ωD-bounded spaces of Lindelöf degree < cov(M) are ω-bounded. \bullet Regular ωD-bounded spaces of countable tightness are ωN-bounded, and if \mathfrakb > ω1 then even ω-bounded. \bullet If a product of Hausdorff space is ωD-bounded then all but one of its factors must be ω-bounded. \bullet Any product of at most \mathfrakt many Hausdorff ωD-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.