  We study the class of Borel equivalence relations under continuous
reducibility. In particular , we characterize when a Borel equivalence relation
with countable equivalence classes is $\Sigma$ 0 $\xi$ (or $\Pi$ 0 $\xi$). We
characterize when all the equivalence classes of such a relation are $\Sigma$ 0
$\xi$ (or $\Pi$ 0 $\xi$). We prove analogous results for the Borel equivalence
relations with countably many equivalence classes. We also completely solve
these two problems for the first two ranks. In order to do this, we prove some
extensions of the Louveau-Saint Raymond theorem which itself generalized the
Hurewicz theorem characterizing when a Borel subset of a Polish space is G
$\delta$ .
