DEGREE SPECTRA OF HOMEOMORPHISM TYPE OF COMPACT POLISH SPACES Article Swipe

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Mathieu Hoyrup , Takayuki Kihara , Victor Selivanov ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.1017/jsl.2023.93 · OA: W4389600084

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 $\mathbf {0}'$ -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 $X_n$ . Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility.