Unboring ideals
Adam Kwela
TL;DR
This work develops a detailed framework for FinBW($\mathcal{I}$), the finite-Bolzano–Weierstrass property relative to ideals on $\omega$, introducing the critical ideal $\mathcal{BI}$ and proving sharp CH/MA–dependent dichotomies: under CH, $\mathcal{BI}\not\leq_K\mathcal{I}$ characterizes the existence of uncountable separable spaces in FinBW($\mathcal{I}$), and $\omega_1$ with the order topology lies in FinBW($\mathcal{I}$) for all $\boldsymbol{\Pi^0_4}$ ideals. The paper also shows that under MA($\sigma$-centered), FinBW($\mathcal{I}$) vs FinBW($\mathcal{J}$) nonemptiness is equivalent to $\mathcal{J}\not\leq_K\mathcal{I}$ for $\boldsymbol{\Pi^0_4}$ ideals, and demonstrates that $\text{Fin} \times \text{Fin}$ is not a $\boldsymbol{\Pi^0_3}$–extendability boundary. In addition, the authors develop a new preorder on ideals, study Mrówka spaces in depth, and apply the theory to Hindman spaces and analytic P-ideals, including simple density and Erdős–Ulam ideals, yielding a rich landscape of FinBW behaviors across Borel classes.
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 $ω$, if for each sequence $(x_n)_{n\inω}$ 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 $\mathcal{I}$, $\mathcal{BI}\not\leq_K\mathcal{I}$ ($\leq_K$ denotes the Katětov preorder of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$). We show that $\mathcal{BI}\not\leq_K\mathcal{I}$ and $ω_1$ with the order topology is in FinBW($\mathcal{I}$), for all $\bf{Π^0_4}$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($σ$-centered) that for $\bf{Π^0_4}$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}\not\leq_K\mathcal{I}$. 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 $\bf{Π^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.