More generalizations of pseudocompactness

Abstract

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 $X$.

22 pages

References in corpus (2)

• More Results on Regular Ultrafilters in ZFC
• Every weakly initially m-compact topological space is mpcap

Cited by in corpus (6)

• Topological spaces compact with respect to a set of filters
• Products of sequentially pseudocompact spaces
• A characterization of the Menger property by means of ultrafilter convergence
• Ordinal Compactness
• Every weakly initially m-compact topological space is mpcap
• Ultrafilter convergence in ordered topological spaces