Some Compactness Properties Related to Pseudocompactness and Ultrafilter Convergence

We discuss some notions of compactness relative to a specified family F of subsets of some topological space X . In particular, we relativize to F the notions ofD-compactness, CAPλ, and [μ, λ]-compactness. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in which F is the family of all singletons of X , in which case we get back the more usual notions. (2) The case in which F is the family of all nonempty open subsets of X , in which case we get notions related to pseudocompactness. Our results concern the mutual relationship among the above compactness properties, and their behavior with respect to products. We generalize to the case of an arbitrary family F some results which are known in particular case (1). Already in particular case (2) most of our results appear to be new. For example, we get characterizations of those spaces which are D-pseudocompact, for some ultrafilter D uniform over λ (Theorem 9). An interesting consequence of our results is that, in case (2), we usually get equivalent notions when we consider closures of open sets, rather than just open sets. No separation axiom is assumed in the present note, unless explicitly mentioned. Suppose that D is an ultrafilter over some cardinal λ, and X is a topological space. 2000 Mathematics Subject Classification. Primary 54A20, 54D20; Secondary 54B10.