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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:
 3e+15 Descriptive set theory 3e+60 Determinacy principles
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References:
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