Comparison of countability conditions within three fundamental classifications of convergences

TL;DR

This paper explores how different countability-based convergence classes relate, using ultrafilter characterizations to connect convergence spaces with classical topology, and provides examples illustrating these relationships.

Contribution

It offers a unified framework linking convergence space classifications with topological properties via ultrafilter characterizations, enriching understanding of their interrelations.

Findings

All classes have ultrafilter characterizations in classical topological terms

Examples from known topological spaces illustrate convergence class distinctions

Ultrafilter approach bridges convergence spaces and topology

Abstract

The interrelations between various classes of convergence spaces defined by countability conditions are studied. Remarkably, they all find characterizations in the usual space of ultrafilters in terms of classical topological properties. This is exploited to produce relevant examples in the realm of convergence spaces from known topological examples.