Canonizing certain Borel equivalences for Silver forcing
type: Item
instance of: scholarly article
title: Canonizing certain Borel equivalences for Silver forcing
zbMATH Open document ID: 1263.03033
author: Michal Doucha
published in: Topology and its Applications
publication date: 2012-08-16T00:00:00Z
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 6066680
zbMATH Keywords: Borel equivalence relation
zbMATH Keywords: Silver ideal
zbMATH Keywords: canonical Ramsey theorem
reviewed by: Szymon Zeberski
MaRDI profile type: Publication
full work available at URL: j.topol.2012.05.008
OpenAlex ID: W2078372635
cites work: Combinatorics on ideals and forcing
cites work: Q3529420
cites work: Canonical Ramsey theory on Polish spaces
cites work: Analytic ideals and their applications
cites work: Q5444715
DOI: 10.1016/J.TOPOL.2012.05.008
review text: 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 \(E_0.\)
description: scientific article; zbMATH DE number 6066680