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A dichotomy for countable unions of smooth Borel equivalence relations (2105.05362v1)
Published 11 May 2021 in math.LO
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.