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New dichotomies for Borel equivalence relations. (English) Zbl 0889.03038

Let \(E\) and \(F\) be Borel equivalence relations on Polish spaces \(X\) and \(Y\), respectively. It is said that \(E\) can be Borel reduced to \(F\), in symbols \(E\leq_B F\), if there is a Borel map \(f:X\to Y\) with \(xEy\Leftrightarrow f(x)Ff(y)\). The paper is a survey of some results and conjectures on the structure of \(\leq_B\), concerning mainly of its initial part. The proved new dichotomies mentioned in the title are two of the previously conjectured dichotomies. One of them is proved only partially. These results simplify the picture of the structure \(\leq_B\) at the first level of \(\leq_B\)-incomparable relations.

MSC:

03E15 Descriptive set theory
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54E50 Complete metric spaces
54H11 Topological groups (topological aspects)
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