Infima and cardinal characteristics of critical ideals for countable compact spaces

TL;DR

This paper investigates the structure and hierarchy of certain ideals related to countable compact spaces, analyzing their order-theoretic properties and computing associated cardinal invariants.

Contribution

It introduces new ideals $ ext{conv}_{<eta}$ to complete the hierarchy and analyzes their complexity and invariants, advancing understanding of these critical ideals.

Findings

$ ext{conv}_eta$ ideals do not serve as greatest lower bounds for limit ordinals

Defined $ ext{conv}_{<eta}$ ideals to fill hierarchy gaps

Computed several cardinal invariants of $ ext{conv}_eta$ ideals

Abstract

For each countable ordinal $\alpha \ge 2$, the ideals $\mathsf{conv}_\alpha$ were introduced in ``Critical ideals for countable compact spaces'' (to appear in Fund. Math., see also arXiv:2503.12571) to characterize compact countable spaces homeomorphic to $\omega^\alpha \cdot n+1$ with the order topology. We study the structure of these ideals in the Kat\v{e}tov order, namely for limit ordinals $\alpha$, we show that $\mathsf{conv}_{\alpha}$ do not serve as greatest lower bounds of the $\mathsf{conv}_\beta$ for $\beta<\alpha$. We therefore define the ideals $\mathsf{conv}_{<\alpha}$ with this property and show that together, the ideals $\mathsf{conv}_\alpha$ and $\mathsf{conv}_{<\alpha}$ form intertwined decreasing hierarchies of $\Sigma^0_4$- and $\Pi^0_5$-complete ideals. Furthermore, we examine several cardinal invariants of $\mathsf{conv}_\alpha$, computing invariants that have recently appeared in the literature in various contexts.