, Volume 89, Issue 1, pp 343356
First online:
Approximation of analytic by Borel sets and definable countable chain conditions
 A. S. KechrisAffiliated withDepartment of Mathematics 25337, California Institute of Technology Email author
 , S. SoleckiAffiliated withDepartment of Mathematics 25337, California Institute of Technology
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LetI be a σideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ _{ 1 } ^{ 1 } countable chain condition if there is aΣ _{ 1 } ^{ 1 } equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ _{ 1 } ^{ 1 } set is equal to a Borel set modulo a set fromI iffI fulfils theΣ _{ 1 } ^{ 1 } countable chain condition. Further we characterize the σidealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ _{ 1 } ^{ 1 } sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σideal of meager sets is the unique σideal onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.
 Title
 Approximation of analytic by Borel sets and definable countable chain conditions
 Journal

Israel Journal of Mathematics
Volume 89, Issue 13 , pp 343356
 Cover Date
 199510
 DOI
 10.1007/BF02808208
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 A. S. Kechris ^{(1)}
 S. Solecki ^{(1)}
 Author Affiliations

 1. Department of Mathematics 25337, California Institute of Technology, 91125, Pasadena, CA, USA