Unboring ideals

Abstract

Our main object of interest is the following notion: 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 $\mathcal{B}\mathcal{I}$ which is critical for FinBW($\mathcal{I}$) in the following sense: Under CH, for every ideal $\mathcal{I}$, $\mathcal{B}\mathcal{I}/{\le }_K\mathcal{I}$ (${\le }_K$ denotes the Katětov preorder of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$). We show that $\mathcal{B}\mathcal{I}/{\le }_K\mathcal{I}$ and ${\omega }_1$ with the order topology is in FinBW($\mathcal{I}$), for all ${\mathbf{\Pi }}_4^0$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($\sigma$-centered) that for ${\mathbf{\Pi }}_4^0$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}/{\le }_K\mathcal{I}$. Moreover, answering in negative a question of M. Hrušák and D. Meza-Alcántara, we show that the ideal $\text{Fin}\times \text{Fin}$ is not critical among Borel ideals for extendability to a ${\mathbf{\Pi }}_3^0$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.

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