Embedding an analytic equivalence relation in the transitive closure of a Borel relation

TL;DR

This paper investigates the complexity of embedding analytic equivalence relations into the transitive closure of Borel relations, showing that any analytic equivalence can be embedded into the union of two smooth Borel relations.

Contribution

It demonstrates that every analytic equivalence relation can be embedded into the transitive closure of two smooth Borel relations, providing a negative answer to a longstanding question.

Findings

Any analytic equivalence relation can be embedded into the transitive closure of two smooth Borel relations.

In the Baire space, the embedding is homeomorphic with the two relations being closed.

The result limits the complexity of classes containing transitive closures of closed relations.

Abstract

The transitive closure of a reflexive, symmetric, analytic relation is an analytic equivalence relation. Does some smaller class contain the transitive closure of every reflexive, symmetric, closed relation? An essentially negative answer is provided here. Every analytic equivalence relation on an arbitrary Polish space is Borel embeddable in the transitive closure of the union of two smooth Borel equivalence relations on that space. In the case of the Baire space, the two smooth relations are closed and the embedding is homeomorphic.