A dichotomy for countable unions of smooth Borel equivalence relations

TL;DR

This paper investigates the structure of countable unions of smooth Borel equivalence relations on Polish spaces, revealing a dichotomy involving Borel reducibility or embedding of a complex relation.

Contribution

It establishes a dichotomy for such unions, showing they are either reducible to countable Borel relations or contain a continuous embedding of E_1, advancing understanding of their complexity.

Findings

Either a Borel reduction to a countable Borel equivalence relation or an embedding of E_1 exists.

Results extend to unions of more general Borel equivalence relations.

Provides a structural classification for unions of smooth Borel equivalence relations.

Abstract

We show that if an equivalence relation $E$ on a Polish space is a countable union of smooth Borel subequivalence relations, then there is either a Borel reduction of $E$ to a countable Borel equivalence relation on a Polish space or a continuous embedding of $\mathbb{E}_1$ into $E$. We also establish related results concerning countable unions of more general Borel equivalence relations.