More generalizations of pseudocompactness

We introduce a covering notion depending on two cardinals, which we call $\mathcal O$-$[ \mu, \lambda ]$-compactness, and which encompasses both pseudocompactness and many other generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to $\mathcal O$-$[ \omega, \omega ]$-compactness. We provide several characterizations of $\mathcal O$-$[ \mu, \lambda ]$-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 $X$.