More generalizations of pseudocompactness

We introduce a covering notion depending on two cardinals, which we call O-[μ,λ]-compactness, and which encompasses both pseudocompactness and many other known 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. The connection turns out to be rather strict when the above notions are considered with respect to products. In passing, we provide some conditions equivalent to D-pseudocompactness.Finally, we show that our methods provide a unified treatment both for O-[μ,λ]-compactness and for [μ,λ]-compactness.