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. vide 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. y, we show that our methods provide a unified treatment both for O - [ μ , λ ] -compactness and for [ μ , λ ] -compactness.