On a Glimm -- Effros dichotomy and an Ulm--type classification in Solovay model
Abstract
We prove that in Solovay model every OD equivalence E on reals either admits an OD reduction to the equality on the set of all countable (of length < omega1) binary sequences, or continuously embeds E0, the Vitali equivalence. If E is a Sigma11 (resp. Sigma12) relation then the reduction in the ``either'' part can be chosen in the class of all Delta1 (resp. Delta2) functions. The proofs are based on a topology generated by OD sets.
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