## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 343-356 |

Number of pages | 14 |

Journal | Israel Journal of Mathematics |

Volume | 89 |

Issue number | 1-3 |

DOIs | |

State | Published - Oct 1995 |

## ASJC Scopus subject areas

- Mathematics(all)