Selection theorems and treeability

Abstract:

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.