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Glimm-Effros for coanalytic equivalence relations. (English) Zbl 1178.03065
The author proves the following result in the style of the Glimm-Effros dichotomy (the latter concerns Borel equivalence relations): If every real has a sharp, then any \(\Pi^1_1\) equivalence relation either Borel reduces \(E_0\) or in a \(\Delta^1_3\) manner allows the assignment of bounded subsets of \(\omega_1\) as complete invariants to the equivalence classes. (The analogous result for \(\Sigma^1_1\) equivalence relations was established earlier by the author and Kechris.) The author asks whether it is possible to replace \(\Delta^1_3\) by \(\Delta^1_2\), and whether the assumption of sharps is necessary.

MSC:
03E15 Descriptive set theory
03E60 Determinacy principles
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