Some compactness properties related to pseudocompactness and ultrafilter convergence

We discuss some notions of compactness and conver- gence relative to a specified family ℱ of subsets of some topological space . The two most interesting cases of our construction appear to be (1) the case in which ℱ is the family of all singletons of , 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 , in which case we get notions related to pseudo- compactness. A large part of the results in this paper are known for case (1); the results are, in general, new in case (2). As an example, we charac- terize those spaces which are -pseudocompact for some ultrafilter uniform over .