A Glimm-Effros dichotomy for Borel equivalence relations
A Glimm-Effros dichotomy for Borel equivalence relations
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DOI：
10.1090/s0894-0347-1990-1057041-5
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发表时间：
1990-01
影响因子：
3.9
通讯作者：
L. Harrington;A. Kechris;A. Louveau
中科院分区：
文献类型：
--
作者：
L. Harrington;A. Kechris;A. Louveau
Glimm [G12]在局部紧群作用的情形下发现了一个关于变换群轨道空间结构的基本二分法，而Effros [E1，E2]在Polish群作用的情形下推广了这个二分法，并且导出的等价关系是Fσ.本文的目的是将Glimm-Effros二分法推广到任意Borel等价关系的一般情形（甚至不一定是由群作用引起的）。尽管我们的结果具有完全经典的描述性集合论性质，但我们的证明需要使用有效的描述性集合论方法，从而最终对整数的可计算性（或递归）理论进行了至关重要的使用。
A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in the Polish group action case when additionally the induced equivalence relation is Fσ. It is the purpose of this paper to extend the Glimm-Effros dichotomy to the very general context of an arbitrary Borel equivalence relation (not even necessarily induced by a group action). Despite the totally classical descriptive set-theoretic nature of our result, our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability (or recursion) theory on the integers.