Dense-set dependence in the Katětov order for uncountable coordinate ideals

Abstract

For each countable ordinal $α\geq 2$, Filipów, Kowalczuk and Kwela introduced an ideal $\mathsf{conv}_α$ on the countable compact ordinal space $ω^α+1$. Kowalczuk later proved that, for each countable limit ordinal $λ$, the ideal $\mathsf{conv}_{<λ}$ is the greatest lower bound of $\{\mathsf{conv}_β:β<λ\}$ in the Katětov order. At the first uncountable level, let $A\subseteq[2,ω_1)$ be uncountable and let $D$ be a countable dense subset of $X_A=\prod_{α\in A}(ω^α+1)$. The coordinate ideal $\mathsf{Conv}(A,D)$ on $D$ consists of those $B\subseteq D$ with $π_α[B]\in\mathsf{conv}_α$ for every $α\in A$. For a pair $D\subseteq E$ of countable dense sets, call $α$ non-small if $π_α[E\setminus D]\notin\mathsf{conv}_α$. In ZFC, if at most countably many coordinates are non-small, then $\mathsf{Conv}(A,D)\equiv_K\mathsf{Conv}(A,E)$. Under CH this countability bound is sharp: for every $A\subseteq[3,ω_1)$ with $|A|=\aleph_1$, there are countable dense sets $D\subseteq D^*\subseteq X_A$ such that $\mathsf{Conv}(A,D^*)\leq_K\mathsf{Conv}(A,D)$ but $\mathsf{Conv}(A,D)\not\leq_K\mathsf{Conv}(A,D^*)$, and in particular $\mathsf{Conv}(A,D)$ and $\mathsf{Conv}(A,D^*)$ are not Katětov equivalent. The non-reduction is obtained, under CH, by diagonalizing along $ω_1$ coordinates against the elements of $ω^ω$ that code retractions $D^*\to D$.