Unboring ideals
Volume 261 / 2023
Abstract
We say that a topological space space $X$ is in FinBW$(\mathcal I)$, where $\mathcal I$ is an ideal on $\omega $, if for each sequence $(x_n)_{n\in \omega}$ in $X$ one can find an $A\notin \mathcal I$ such that $(x_n)_{n\in A}$ converges in $X$.
We define an ideal $\mathcal{BI}$ which is critical for FinBW$(\mathcal I)$ in the following sense: Under CH, for every ideal $\mathcal I$, $\mathcal{BI}\nleq _{\rm K}\mathcal I$ ($\leq _{\rm K}$ denotes the Katětov preorder of ideals) iff there is an uncountable separable space in FinBW$(\mathcal I)$. We show that $\mathcal{BI}\nleq _{\rm K}\mathcal I$ and $\omega _1$ with the order topology is in FinBW$(\mathcal I)$ for all $\mathbf{\Pi}^0_4$ ideals $\mathcal I$.
We examine when $\mathrm{FinBW}(\mathcal I)\setminus \mathrm{FinBW}(\mathcal J)$ is nonempty: under MA($\sigma $-centered) we prove that for $\mathbf{\Pi}^0_4$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}\nleq _{\rm K}\mathcal{I}$. Moreover, answering in the negative a question of M. Hrušák and D. Meza-Alcántara, we show that the ideal $\mathrm{Fin} \times \mathrm{Fin}$ is not critical among Borel ideals for extendability to a $\mathbf{\Pi}^0_3$ ideal. Finally, we apply our results to the study of Hindman spaces and in the context of analytic P-ideals.
