Infima and cardinal characteristics of critical ideals for countable compact spaces

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 Katětov 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 ${\Sigma }_4^0$- and ${\Pi }_5^0$-complete ideals. Furthermore, we examine several cardinal invariants of ${\mathsf{conv}}_{\alpha }$, computing invariants that have recently appeared in the literature in various contexts.