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Approximation of analytic by Borel sets and definable countable chain conditions

Kechris, A. S. and Solecki, S. (1995) Approximation of analytic by Borel sets and definable countable chain conditions. Israel Journal of Mathematics, 89 (1-3). pp. 343-356. ISSN 0021-2172. doi:10.1007/BF02808208.

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Let I be a σ-ideal on a Polish space such that each set from I is contained in a Borel set from I. We say that I fails to fulfil the Σ_1^1 countable chain condition if there is a Σ_1^1 equivalence relation with uncountably many equivalence classes none of which is in I. Assuming definable determinacy, we show that if the family of Borel sets from I is definable in the codes of Borel sets, then each Σ_1^1 set is equal to a Borel set modulo a set from I iff I fulfils the Σ_1^1 countable chain condition. Further we characterize the σ-ideals I generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property for Σ_1^1 sets mentioned above. It turns out that they are exactly of the form MGR(F)={A : ∀F ∈ F A ∩F is meager in F} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal on R, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.

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Additional Information:© 1995 Springer-Verlag. Received July 20, 1993 and in revised form March 6, 1994. Research partially supported by NSF grant DMS-9317509.
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Issue or Number:1-3
Record Number:CaltechAUTHORS:20130521-103730289
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Official Citation:Approximation of analytic by Borel sets and definable countable chain conditions A. S. Kechris, S. Solecki pp.343-356
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38600
Deposited By: Ruth Sustaita
Deposited On:21 May 2013 17:54
Last Modified:09 Nov 2021 23:38

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