We introduce a covering notion depending on two cardinals, which we call
O-[μ,λ]-compactness, and which encompasses both
pseudocompactness and many other generalizations of pseudocompactness. For
Tychonoff spaces, pseudocompactness turns out to be equivalent to O-[ω,ω]-compactness. We provide several characterizations of
O-[μ,λ]-compactness, and we discuss its connection
with D-pseudocompactness, for D an ultrafilter. We analyze the behaviour of
the above notions with respect to products. Finally, we show that our results
hold in a more general framework, in which compactness properties are defined
relative to an arbitrary family of subsets of some topological space X.Comment: 22 page