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Journal Articles The Journal of Symbolic Logic Year : 2023

Degree Spectra of Homeomorphism Types of Compact Polish Spaces

Abstract

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a 0'-computable low_3 compact Polish space which is not homeomorphic to a computable one, and that, for any natural number n\geq 2, there exists a Polish space X_n such that exactly the high_n-degrees are required to present the homeomorphism type of X_n. We also show that no compact Polish space has a least presentation with respect to Turing reducibility. The first version of this article appeared in April 2020. A major update was made in September 2023, with improved proofs and results. This is the final version from January 2024, with more results on Čech homology groups.
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Dates and versions

hal-02555111 , version 1 (27-04-2020)
hal-02555111 , version 2 (18-09-2023)
hal-02555111 , version 3 (02-01-2024)

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Mathieu Hoyrup, Takayuki Kihara, Victor Selivanov. Degree Spectra of Homeomorphism Types of Compact Polish Spaces. The Journal of Symbolic Logic, In press, pp.1-32. ⟨10.1017/jsl.2023.93⟩. ⟨hal-02555111v3⟩
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