On a Glimm -- Effros dichotomy theorem for Souslin relations in generic universes

Abstract

We prove that if every real belongs to a set generic extension of the constructible universe then every Σ_1^1 equivalence E on reals either admits a Delta_1^HC reduction to the equality on the set 2^{<\om_1} of all countable binary sequences, or continuously embeds E_0, the Vitali equivalence. The proofs are based on a topology generated by OD sets.