DEGREE SPECTRA OF HOMEOMORPHISM TYPE OF COMPACT POLISH SPACES
Mathieu Hoyrup, Takayuki Kihara, Victor Selivanov · Journal of Symbolic Logic · 2023
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 bold 0 prime $\mathbf {0}'$ 0 ' -computable low Subscript 3 $_3$ 3 compact Polish space which is not homeomorphic to a computable one, and that, for any natural number n greater than or equals 2 $n\geq 2$ n ≥ 2 , there exists a Polish space upper X Subscript n $X_n$ X n such that exactly the high Subscript n $_{n}$ n -degrees are required to present the homeomorphism type of upper X Subscript n $X_n$ 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.