On a conjecture of Debs and Saint Raymond
Volume 260 / 2023
Fundamenta Mathematicae 260 (2023), 67-76
MSC: Primary 03E05; Secondary 03E15, 54H05, 26A03.
DOI: 10.4064/fm111-5-2022
Published online: 8 September 2022
Abstract
The Borel separation rank of an analytic ideal $\mathcal {I}$ on $\omega $ is the minimal ordinal $\alpha \lt \omega _{1}$ such that there is $\mathcal {S}\in \boldsymbol\Sigma ^0_{1+\alpha }$ with $\mathcal I\subseteq \mathcal S$ and $\mathcal {I}^\star \cap \mathcal {S}=\emptyset $, where $\mathcal I^\star $ is the filter dual to the ideal $\mathcal I$. Answering in negative a question of G. Debs and J. Saint Raymond [Fund. Math. 204 (2009)], we construct a Borel ideal of rank $ \gt 2$ which does not contain an isomorphic copy of the ideal $\text {Fin}^3$.
