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 connec- tion 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 pro- vide 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.