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

On the complexity of Borel equivalence relations with some countability property