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DEGREE SPECTRA OF HOMEOMORPHISM TYPE OF COMPACT POLISH SPACES
Published online by Cambridge University Press: 11 December 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 $\mathbf {0}'$-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.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
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