On the complexity of Borel equivalence relations with some countability property

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 $\mathbf {\Sigma }^{0}_{\xi }$ (or $\mathbf {\Pi }^{0}_{\xi }$). We characterize when all the equivalence classes of such a relation are $\mathbf {\Sigma }^{0}_{\xi }$ (or $\mathbf {\Pi }^{0}_{\xi }$). 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 $G_\delta$.