Remark on meager \(\Sigma_2^0\)-supported \(\sigma \)-ideals on the real line (Q2774516)

An ideal \(\mathcal I\) on \(\mathbb R\) is \(\Sigma ^0_2\)-supported if each set \(A\in \mathcal I\) is contained in an \(F_\sigma \) set \(B\in \mathcal I\). The authors consider this statement: For each \(\Sigma ^0_2\)-supported \(\sigma \)-ideal \(\mathcal I\subset \mathcal P(\mathbb R)\) containing all singletons such that \(({\forall A\in \mathcal I}) \allowbreak ({\exists B\in [\mathbb R\smallsetminus A]^{\mathfrak c}}) \allowbreak ({B\in \mathcal I})\) there exists a Polish topology \(\tau \) on \(\mathbb R\) such that \(\mathcal I\) is the family of meager sets in the topology \(\tau \). They prove the independence of the statement (it follows from CH while MA\({}+\neg {}\)CH implies the negation of it). In fact, the authors show that the validity of the statement partially depends on cardinal invariants of the ideal of meager sets (and of the ideal \(\mathcal I\)) and the existence of a Luzin set.