Critical partition regular functions for compact spaces

2026/01/17 by Rafał Filipów, Małgorzata Kowalczuk, Hubert Książek +2 · 1 voice
Mathematics · #math.GN

paper · pdf · doi:10.1016/j.topol.2026.109804

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.

Citations

• A unified approach to Hindman, Ramsey and van der Waerden spaces
• Combinatorial properties of MAD families
• Unboring ideals
• Strongly mixing systems are almost strongly mixing of all orders
• Van der Waerden spaces and Hindman spaces are not the same

Discussions

• Rafa{\l} Filip\'ow, Ma{\l}gorzata Kowalczuk, Hubert Ksi\k{a}\.zek, Adam Kwela, Grzegorz Ucal: Critical partition regular functions for compact spaces https://arxiv.org/abs/2601.12041 https://arxiv.org [bsky, 0 points, 0 comments]

Related

• A new order for ideal sequential compactness
• Critical ideals for compact spaces
• Compactness with ideals
• Sets of Ramsey-limit points and IP-limit points
• Unboring ideals