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

Abstract

For each countable ordinal $\alpha \ge 2$, Filipów, Kowalczuk and Kwela introduced an ideal ${\mathsf{conv}}_{\alpha }$ on the countable compact ordinal space ${\omega }^{\alpha }+1$. Kowalczuk later proved that, for each countable limit ordinal $\lambda$, the ideal ${\mathsf{conv}}_{<\lambda }$ is the greatest lower bound of $\{{\mathsf{conv}}_{\beta }:\beta <\lambda \}$ in the Katětov order. At the first uncountable level, let $A\subseteq [2,{\omega }_1)$ be uncountable and let $D$ be a countable dense subset of $X_A={\prod }_{\alpha \in A}({\omega }^{\alpha }+1)$. The coordinate ideal $\mathsf{Conv}(A,D)$ on $D$ consists of those $B\subseteq D$ with ${\pi }_{\alpha }[B]\in {\mathsf{conv}}_{\alpha }$ for every $\alpha \in A$. For a pair $D\subseteq E$ of countable dense sets, call $\alpha$ non-small if ${\pi }_{\alpha }[E\setminus D]\notin {\mathsf{conv}}_{\alpha }$. 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 $A\subseteq [3,{\omega }_1)$ with $|A|={\aleph }_1$, there are countable dense sets $D\subseteq D^{\ast }\subseteq X_A$ such that $\mathsf{Conv}(A,D^{\ast }){\le }_K\mathsf{Conv}(A,D)$ but $\mathsf{Conv}(A,D)/{\le }_K\mathsf{Conv}(A,D^{\ast })$, and in particular $\mathsf{Conv}(A,D)$ and $\mathsf{Conv}(A,D^{\ast })$ are not Katětov equivalent. The non-reduction is obtained, under CH, by diagonalizing along ${\omega }_1$ coordinates against the elements of ${\omega }^{\omega }$ that code retractions $D^{\ast }\to D$.

14 pages