Canonizing certain Borel equivalences for Silver forcing
The author studies Borel equivalence relations with connection to Silver forcing.NEWLINENEWLINEHe proves the following result. Assume that if \(E\subseteq B^2\) is a Borel equivalence relation, where \(B\) is a condition in the Silver forcing, and \(E\leq_B E_2\), i.e., there is a Borel function \(f:B\rightarrow 2^\omega\) such that \((\forall x,y\in B)(xEy\leftrightarrow \sum\left\{\frac{1}{n+1}:\;f(x)(n)\neq f(y)(n)\right\}<\infty)\). Then there is a Silver condition \(C\subseteq B\) such that one of the following holds:NEWLINENEWLINE1. \(E\upharpoonright C=C^2\),NEWLINENEWLINE2. \(E\upharpoonright C\) is a subset of \(E_0\) (\(xE_0 y\leftrightarrow \{n:\;x(n)\neq y(n)\}\) is finite).NEWLINENEWLINEMoreover, the same conclusion is true for \(E=E_{\mathcal I}\) for \(F_\sigma\) P-ideal \(\mathcal I\) and for \(E=l^p\) for \(p\in [1,\infty)\), where \(x l^p y\leftrightarrow x-y\in l^p\).NEWLINENEWLINEOn the other hand, the conclusion does not hold for \(E=l^{\infty}\).NEWLINENEWLINEIn the second part of the paper, the author shows that there are perfectly many essentially different non-homogenous subequivalences of \(E_0.\)
scientific article; zbMATH DE number 6066680