More generalizations of pseudocompactness

TL;DR

This paper introduces a new generalization of pseudocompactness called $ ext{O}$-$[ ext{mu}, ext{lambda}]$-compactness, providing characterizations, product behavior, and a broader framework for compactness properties.

Contribution

It defines a new covering notion depending on two cardinals that unifies pseudocompactness and other generalizations, with characterizations and product analysis.

Findings

$ ext{O}$-$[ ext{omega}, ext{omega}]$-compactness is equivalent to pseudocompactness for Tychonoff spaces

Provides multiple characterizations of the new compactness notion

Analyzes behavior of the notion under products and in a generalized framework

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 $X$.