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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 $delta$ .
Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibil
The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations $R$ and $S$ on natural numbers, $R$ is computably reducible to $
We examine the degree structure $mathbf{ER}$ of equivalence relations on $omega$ under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not have a join bu
We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems
We generalise the main theorems from the paper The Borel cardinality of Lascar strong types by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental