We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

中文翻译：

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图灵不变集和完美集属性

我们证明$ZF+DC+$“所有图灵不变实数集都具有完美集属性”意味着所有实数集都具有完美集属性。我们还表明，这个结果可以推广到所有可数的解析等价关系。