  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.
