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 < omega_1) binary sequences, or continuously embeds E_0, the Vitali equivalence. If E is a Sigma_1^1 (resp. Sigma_1^2) relation then the reduction in the ``either&#39;&#39; part can be chosen in the class of all Delta_1 (resp. Delta_2) functions. The proofs are based on a topology generated by OD sets.