We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts, which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.

中文翻译：

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几类COMPACTA的同胚关系的复杂性

我们证明了紧致空间之间的同胚关系可以不断地简化为绝对缩进之间的同胚等价关系，这加强和简化了Chang和Gao以及Cieśla最近的结果。由此可见，绝对回缩的同胚关系是与万能轨道等价关系的Borel双可约关系。我们还证明了正则连续体之间的同胚关系可以被可数结构分类，因此它与可数集上置换群的全域轨道等价关系是Borel双可约的。另一方面，我们证明了边缘有限可度量致密体之间的同胚关系不能被可数结构分类。