Selection theorems and treeability

We show that domains of non-trivial \Sigma ^1_1 trees have \Delta ^1_1 members. Using this, we show that smooth treeable equivalence relations have Borel transversals, and essentially countable treeable equivalence relations have Borel complete countable sections. We show also that treeable equivale...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2008-10, Vol.136 (10), p.3647-3653
Main Author: Hjorth, Greg
Format: Article
Language:English
Subjects:
Descriptive set theory
Equivalence relation
Logical theorems
Mathematical theorems
Mathematics
Same sex marriage
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Summary:We show that domains of non-trivial \Sigma ^1_1 trees have \Delta ^1_1 members. Using this, we show that smooth treeable equivalence relations have Borel transversals, and essentially countable treeable equivalence relations have Borel complete countable sections. We show also that treeable equivalence relations which are ccc idealistic, measured, or generated by a Borel action of a Polish group have Borel complete countable sections.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09548-8