A dichotomy for countable unions of smooth Borel equivalence relations

TL;DR

The paper proves a dichotomy for $\sigma$-smooth Borel equivalence relations on Polish spaces: such an $E$ is either Borel reducible to a countable Borel equivalence relation or contains a continuous copy of $\mathbb{E}_1$. It develops two technical dichotomies via aligned mappings and cores, extending the Kechris–Louveau framework to unions of subequivalence relations and to broader classes like strongly-idealistic and potentially-$F_\sigma$ relations. The results unify and extend existing dichotomies, showing that any $E$ that is a countable union of subrelations reducible to a robust class $\mathcal{F}$ is either reducible to an $F\in\mathcal{F}$ or admits an embedding of $\mathbb{E}_1$, with exclusivity in the ccc-idealistic case. These theorems have implications for orbit equivalence relations and hyperfinite/hypersmooth hierarchies, informing how complex Borel relations can be decomposed or embedded into canonical benchmarks like $\mathbb{E}_1$.

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.

Theorems & Definitions (99)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Proposition 2.1
• proof
• Proposition 2.2: see Engelking
• Proposition 2.3