Generalized Luzin sets (Q2862658)

The authors introduce the notion of a generalized \((\mathcal{I},\mathcal{J})\)-Luzin set. Let \(\mathcal{X}\) denote an uncountable Polish space. Let \(\mathcal{I}\), \(\mathcal{J}\) denote \(\sigma\)-ideals on \(\mathcal{X}\) each of which has a Borel base and \([\mathcal{X}]^{\leq\omega}\subseteq\mathcal{I},\mathcal{J}\). A subset \(L\) of \(\mathcal{X}\) is said to be:NEWLINENEWLINE-- {a \((\mathcal{I},\mathcal{J})\)-Luzin set} if \(L\notin\mathcal{I}\) but \((\forall B\in\mathcal{I})(B\cap L\in\mathcal{J})\);NEWLINENEWLINE-- {a \((\kappa,\mathcal{I},\mathcal{J})\)-Luzin set}, where \(\kappa\) is a cardinal, if \(L\) is a \((\mathcal{I},\mathcal{J})\)-Luzin set and \(|L|=\kappa\).NEWLINENEWLINENote that \(L\) is a Luzin set iff \(L\) is a generalized \((\mathbb{L},[\mathbb{R}]^{\leq\omega})\)-Luzin set and \(S\) is a Sierpiński set iff \(S\) is a generalized \((\mathbb{K},[\mathbb{R}]^{\leq\omega})\)-Luzin set. Here, \(\mathbb{L}\) denotes the ideal of Lebesgue measure zero sets and \(\mathbb{K}\) denotes the ideal of meager sets.NEWLINENEWLINERecall that if \(\mathcal{F}\) is a family of functions in \(^{\mathcal{X}}\mathcal{X}\) and \(A,B\subseteq\mathcal{X}\), than \(A\) and \(B\) are said to be equivalent with respect to \(\mathcal{F}\) if there is \(f\in\mathcal{F}\) such that \(B=f[A]\) or \(A=f[B]\). Generalizing the Erdős-Sierpiński duality theorem (see [\textit{J. C. Oxtoby}, Measure and category. A survey of the analogies between topological and measure spaces. 2nd ed. Berlin: Springer-Verlag (1980; Zbl 0435.28011)]), the authors show that if \(\kappa=\text{cov}(\mathcal{I})=\text{cof}(\mathcal{I})\leq\text{non}(\mathcal{J})\) and \(\mathcal{F}\subseteq {^\mathcal{X}\mathcal{X}}\) is a family of cardinality \(\leq\kappa\), then there is a sequence \((L_\alpha)_{\alpha<\kappa}\) of \((\kappa,\mathcal{I},\mathcal{J})\)-Luzin sets, which are not pairwise equivalent with respect to \(\mathcal{F}\). The theorem implies among others that if \(2^\omega=\text{cov}(\mathcal{I})=\text{non}(\mathcal{J})\) then there are continuum many different \((\mathcal{I},\mathcal{J})\)-Luzin sets which are not Borel equivalent.NEWLINENEWLINEIt is observed that every Sierpiński set (resp. Luzin set) remains a Sierpiński set (resp. Luzin set) in the random (resp. Cohen) forcing extension. In addition, the paper provides a number of sufficient conditions for a forcing notion \(\mathbb{P}\) to preserve \((\mathcal{I},\mathcal{J})\)-Luzin sets.