Dense-set dependence in the Kat\v{e}tov order for uncountable coordinate ideals

Abstract

For each countable ordinal $\alpha\geq 2$, Filip\'ow, 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 Kat\v{e}tov 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 $D\subseteq D^*\subseteq X_A$ 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 Kat\v{e}tov equivalent. The non-reduction is obtained, under CH, by diagonalizing along $\omega_1$ coordinates against the elements of $\omega^\omega$ that code retractions $D^*\to D$.