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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2004.06872 |
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Table of Contents:
- 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.