Critical partition regular functions for compact spaces

TL;DR

This paper explores ideal-based refinements of sequential compactness in topological spaces, introducing critical ideals and partition regular functions that unify classical and non-classical convergence notions.

Contribution

It extends the theory of critical ideals to a broader framework using partition regular functions, linking classical and non-classical convergence concepts.

Findings

Characterization of classical topological classes via critical ideals

Introduction of a unifying framework for various convergence notions

Analysis of critical ideals related to Mazurkiewicz's theorem

Abstract

We study ideal-based refinements of sequential compactness arising from the class FinBW(I), consisting of topological spaces in which every sequence admits a convergent subsequence indexed by a set outside a given ideal I. A central theme of this work is the existence of critical ideals whose position in the Katetov order determines the relationship between a fixed class of spaces and the corresponding FinBW(I) classes. Building on earlier results characterizing several classical topological classes via such ideals, we extend this theory to a broader framework based on partition regular functions, which unifies ordinary convergence with other non-classical convergence notions such as IP- and Ramsey-type convergence. Furthermore, we investigate the existence of critical ideals associated with function classes motivated by Mazurkiewicz's theorem on uniformly convergent subsequences.