A Separation Result for Countable Unions of Borel Rectangles

Lecomte, Dominique

doi:10.1017/jsl.2019.13

J. symb. log.

2019

DOI: 10.1017/jsl.2019.13

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A Separation Result for Countable Unions of Borel Rectangles

Dominique Lecomte¹

Abstract: We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of Σ 0 1 ×Σ 0 ξ sets, or by a Π 0 1 ×Π 0 ξ set.

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On the complexity of Borel equivalence relations with some countability property

Lecomte

2019

Trans. Amer. Math. Soc.

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We study the class of Borel equivalence relations under continuous reducibility. In particular, we characterize when a Borel equivalence relation with countable equivalence classes is Σ 0 ξ (or Π 0 ξ ). We characterize when all the equivalence classes of such a relation are Σ 0 ξ (or Π 0 ξ ). We prove analogous results for the Borel equivalence relations with countably many equivalence classes. We also completely solve these two problems for the first two ranks. In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewicz theorem characterizing when a Borel subset of a Polish space is G δ .

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On the complexity of Borel equivalence relations with some countability property

Lecomte

2019

Trans. Amer. Math. Soc.

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A Separation Result for Countable Unions of Borel Rectangles

Abstract: We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of Σ 0 1 ×Σ 0 ξ sets, or by a Π 0 1 ×Π 0 ξ set.

Cited by 1 publication

References 14 publications

On the complexity of Borel equivalence relations with some countability property

On the complexity of Borel equivalence relations with some countability property

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