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Infima and cardinal characteristics of critical ideals for countable compact spaces

Malgorzata Kowalczuk

Abstract

For each countable ordinal $α\ge 2$, the ideals $\mathsf{conv}_α$ 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 $ω^α\cdot n+1$ with the order topology. We study the structure of these ideals in the Katětov order, namely for limit ordinals $α$, we show that $\mathsf{conv}_α$ do not serve as greatest lower bounds of the $\mathsf{conv}_β$ for $β<α$. We therefore define the ideals $\mathsf{conv}_{<α}$ with this property and show that together, the ideals $\mathsf{conv}_α$ and $\mathsf{conv}_{<α}$ form intertwined decreasing hierarchies of $Σ^0_4$- and $Π^0_5$-complete ideals. Furthermore, we examine several cardinal invariants of $\mathsf{conv}_α$, computing invariants that have recently appeared in the literature in various contexts.

Infima and cardinal characteristics of critical ideals for countable compact spaces

Abstract

For each countable ordinal , the ideals 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 with the order topology. We study the structure of these ideals in the Katětov order, namely for limit ordinals , we show that do not serve as greatest lower bounds of the for . We therefore define the ideals with this property and show that together, the ideals and form intertwined decreasing hierarchies of - and -complete ideals. Furthermore, we examine several cardinal invariants of , computing invariants that have recently appeared in the literature in various contexts.
Paper Structure (4 sections, 11 theorems, 37 equations, 1 figure)

This paper contains 4 sections, 11 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Infima of critical ideals for countable compact spaces
  4. Cardinal characteristics of critical ideals for countable compact spaces

Key Result

Proposition 2.3

Let $A\subseteq \omega^{\alpha}+1$. The following conditions are equivalent.

Figures (1)

  • Figure 1: Relationships between defined cardinal characteristics for tall ideals ($\kappa\to\lambda$ means $\kappa\leq \lambda$ in this diagram).

Theorems & Definitions (25)

  • Example 2.1
  • Definition 2.2
  • Proposition 2.3: critical-ideals-RF-MK-AK
  • Proposition 2.4: critical-ideals-RF-MK-AK
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Claim
  • proof : Proof of Claim
  • Proposition 3.3
  • ...and 15 more