Some compactness properties related to pseudocompactness and ultrafilter convergence

TL;DR

This paper explores generalized notions of compactness and convergence in topological spaces relative to a family of subsets, focusing on cases related to pseudocompactness and ultrafilter convergence, providing new characterizations.

Contribution

It introduces new notions of compactness based on families of subsets, especially for pseudocompactness, and characterizes D-pseudocompact spaces using ultrafilters.

Findings

Characterization of D-pseudocompact spaces

New results for pseudocompactness related to ultrafilter convergence

Extension of classical compactness notions

Abstract

We discuss some notions of compactness and convergence relative to a specified family F of subsets of some topological space X. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in which F is the family of all singletons of X, in which case we get back the more usual notions. (2) The case in which F is the family of all nonempty open subsets of X, in which case we get notions related to pseudocompactness. A large part of the results in this note are known in particular case (1); the results are, in general, new in case (2). As an example, we characterize those spaces which are D-pseudocompact, for some ultrafilter D uniform over $\lambda$.