On Farah ideals and beyond

Abstract

An ideal $\mathcal{I}$ on $ω$ is called Farah if there are hereditary closed subsets $C_n$ of $\mathcal{P}(ω)$, for $n\inω$, such that $\mathcal{I}=\{A\subseteqω:\ \forall_{n\inω}\ \exists_{k\inω}\ A\setminus k\in C_n\}$. Farah asked in [Trans. Amer. Math. Soc. 356 (2004), no. 6] if every $\mathbf{F_{σδ}}$ ideal is a Farah ideal. We show that this is false by providing both tall and non-tall counterexamples. Moreover, we show that there is an antichain in the sense of Katětov order of length $\continuum$ consisting of $\mathbf{F_{σδ}}$ non-Farah ideals. We also investigate a closely related problem concerning sets of ideal limit points. We show that if $\mathcal{I}$ is an $\mathbf{G_{δσδ}}$ ideal on $ω$ and $x\in X^ω$ is a sequence in an uncountable Polish space $X$, then \[ Λ_x(\mathcal{I})=\left\{p\in X:\ \exists_{A\notin\mathcal{I}}\ x\restriction A\text{ converges to }p\right\} \] is $\mathbf{F_σ}$. This answers [Top. App. 312 (2022)] (partially) and [J. Symb. Logic, DOI: 10.1017/jsl.2026.10181]. Earlier it was known to be true only in the case of Farah ideals.