Critical ideals for compact spaces
Rafał Filipów, Małgorzata Kowalczuk, Adam Kwela
TL;DR
The paper develops a fine-grained ideal-theoretic framework to classify compact countable spaces via the Mazurkiewicz–Sierpiński description, defining convexity-based critical ideals conv_α and linking convergence patterns to the FinBW classes through Katětov order. It proves that conv_{1+α} ot a_K \\mathcal{I} is equivalent to $ω^α+1 ∈ FinBW(\mathcal{I})$, yielding a precise correspondence FinBW(\mathcal{I}) ∩ 𝕂 = 𝕂_α; additionally, conv_β ≤_K conv_α for α<β, establishing a strictly decreasing chain in the Katětov order. The work also analyzes the Katětov and Borel properties of conv_α, shows ω_1 cannot be captured by a single critical ideal, and connects these ideals to classic cardinal characteristics, thereby offering a robust, multi-faceted framework for understanding convergence patterns on compact countable spaces. These results deepen the link between topology of countable compacta and set-theoretic structure of tall ideals, with implications for ordinal-based classifications and descriptive set theory.
Abstract
For each countable ordinal $α$, we introduce an ideal $conv_α$ and use it to characterize the class of all compact countable spaces which are homeomorphic to the space $ω^α\cdot n+1$ with the order topology. The characterization is expressed in terms of finding a convergent subsequence defined on a set not belonging to $conv_α$.