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.


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)
Full Text: DOI Link Link


[1] DOI: 10.1090/S0002-9947-1994-1149121-0
[2] The descriptive set theory of Polish group actions 232 (1996) · Zbl 0949.54052
[3] Admissible sets and structures (1975)
[4] Journal 2 pp 339– (1996)
[5] DOI: 10.1016/0003-4843(80)90002-9 · Zbl 0517.03018
[6] DOI: 10.1090/S0002-9939-1994-1169042-2
[7] DOI: 10.1090/S0002-9947-1977-0578656-4
[8] DOI: 10.1090/S0894-0347-97-00221-X · Zbl 0865.03039
[9] Classical descriptive set theory 156 (1995) · Zbl 0819.04002
[10] Ergodic Theory and Dynamical Systems 12 pp 283– (1992)
[11] DOI: 10.2307/2274913 · Zbl 0734.03028
[12] DOI: 10.1016/S0168-0072(96)00006-1 · Zbl 0933.03056
[13] Journal of the American Mathematical Society pp 903– (1990)
[14] Studies in logic and the foundations of mathematics 134 pp 151– (1994)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.