Splitting squares
type: Item
instance of: scholarly article
title: Splitting squares
zbMATH Open document ID: 1144.03029
author: Otmar Spinas
published in: Israel Journal of Mathematics
publication date: 2008-05-27T00:00:00Z
review text: A set \(A\subseteq2^\omega\) is a splitting family if for every \(a\in[\omega]^\omega\) there exists \(x\in A\) such that \(\exists^\infty n\in a\) \(x(n)=0\) and \(\exists^\infty n\in a\) \(x(n)=1\), i.e., \(x\) splits \(a\). The set \(A\) is called countably splitting, if for every countable set \(F\subseteq[\omega]^\omega\) we can find \(x\in A\) which splits every element of \(F\). The author proves that for every Borel function \(f:[2^\omega]^n\to2^\omega\) there exists a closed countably splitting family \(A\) such that \(f\restriction[A]^n\) omits a perfect set of values in \(2^\omega\). This result answers in the negative a question of J. Zapletal, and it is somewhat stronger than the question requires.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E02
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E40
zbMATH DE Number: 5279133
zbMATH Keywords: coutably splitting family
zbMATH Keywords: partition calculus
zbMATH Keywords: lattice of constructibility degrees
zbMATH Keywords: tree forcing
MaRDI profile type: Publication
full work available at URL: s11856-007-0087-6
OpenAlex ID: W3188779758
cites work: A Combinatorial Theorem
cites work: Q4323294
cites work: Canonical forms of Borel functions on the Milliken space
cites work: Q4694312
cites work: Canonical behavior of Borel functions on superperfect rectangles
cites work: Analytic countably splitting families
cites work: Complexity of reals in inner models of set theory
DOI: 10.1007/S11856-007-0087-6
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description: scientific article; zbMATH DE number 5279133