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. This version is a major update from September 2023, with improved proofs and results.
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