Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Degree spectra of homeomorphism types of Polish spaces   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 Polish spaces. We show that there exists a
$0'$-computable low$_3$ Polish space which is not homeomorphic to a computable
one, and that, for any natural number $n$, there exists a Polish space $X_n$
such that exactly the high$_{2n+3}$-degrees are required to present the
homeomorphism type of $X_n$. We also show that no compact Polish space has an
easiest presentation with respect to Turing reducibility.

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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th