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'' 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.

References in corpus (2)

• Descriptive set theory and forcing; How to prove theorems about Borel sets the hard way
• On a Glimm -- Effros dichotomy theorem for Souslin relations in generic universes

Cited by in corpus (10)

• Countable OD sets of reals belong to the ground model
• In Cohen generic extension, every countable OD set of reals belongs to the ground model
• OD elements of countable OD sets in the Solovay model
• On a Glimm -- Effros dichotomy theorem for Souslin relations in generic universes
• On effective sigma-boundedness and sigma-compactness
• Linearization of analytic order relations
• Linearization of partial quasi-orderings in the Solovay model revisited
• Bounding and decomposing thin analytic partial orderings
• On countable cofinality of definable chains in Borel partial orders
• On countable cofinality and decomposition of definable thin orderings