Critical partition regular functions for compact spaces
Rafał Filipów, Małgorzata Kowalczuk, Hubert Książek, Adam Kwela, Grzegorz Ucal
TL;DR
The paper develops a unified framework for ideal-based refinements of convergence by introducing partition regular functions and studying their FinBW properties within the Katětov-order landscape. It generalizes classical results by relating FinBW$(\mathcal{I})$ to topological classes (finite, boring, metric compact) and by identifying critical rho's via reductions in the Katětov order; it further constructs new partition regular functions not arising from any ideal and proves an ideal version of Mazurkiewicz’s theorem, showing FinBW properties control uniform subsequences on perfect sets. A key technical achievement is showing how metric-compact spaces align with convergence notions through a reduction to rational-valued encodings, enabling precise criteria such as $[0,1] \in \mathrm{FinBW}(\rho)$ being equivalent to $\rho_{\mathrm{conv}} \nleq_K \rho$. Overall, the work provides a cohesive, extensible approach to comparing non-classical convergence notions (IP-, Ramsey-type) within a Katětov-order framework, with concrete characterizations for fundamental space classes and new avenues for identifying critical ideals and partition-regular function classes.
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.