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DEGREE SPECTRA OF HOMEOMORPHISM TYPE OF COMPACT POLISH SPACES
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 11 December 2023, pp. 1-32
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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.
THE SIMPLEST LOW LINEAR ORDER WITH NO COMPUTABLE COPIES
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 1 / March 2024
- Published online by Cambridge University Press:
- 10 June 2022, pp. 97-111
- Print publication:
- March 2024
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A low linear order with no computable copy constructed by C. Jockusch and R. Soare has Hausdorff rank equal to
$2$. In this regard, the question arises, how simple can be a low linear order with no computable copy from the point of view of the linear order type? The main result of this work is an example of a low strong 
$\eta $-representation with no computable copy that is the simplest possible example.
