Israel Journal of Mathematics

, Volume 89, Issue 1, pp 343–356

Approximation of analytic by Borel sets and definable countable chain conditions


DOI: 10.1007/BF02808208

Cite this article as:
Kechris, A.S. & Solecki, S. Israel J. Math. (1995) 89: 343. doi:10.1007/BF02808208


LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ11 countable chain condition if there is aΣ11 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 eachΣ11 set is equal to a Borel set modulo a set fromI iffI fulfils theΣ11 countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ11 sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀FFAF is meager inF} 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 onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.

Copyright information

© Hebrew University 1995

Authors and Affiliations

  1. 1.Department of Mathematics 253-37California Institute of TechnologyPasadenaUSA