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arXiv · 2508.06187

A non-computable c.e. closed subset of $[0,1]$

Serikzhan Badaev·Nikolay Bazhenov·Sergey Goncharov·Birzhan Kalmurzayev·Alexander Melnikov
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Abstract

We prove that there exists a $\Sigma^0_1$ closed subset of $[0,1]$ that is not homeomorphic to any computably compact space. We show that the index set of c.e. subspaces of $[0,1]$ that admit a computably compact presentation is not arithmetical, as witnessed by subsets of $[0,1]$. The index set result is new for computable Polish spaces in general, not only for those realised as c.e. closed subsets of $[0,1]$.

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Serikzhan Badaev, Nikolay Bazhenov, Sergey Goncharov, Birzhan Kalmurzayev, Alexander Melnikov. 2025-08-08. A non-computable c.e. closed subset of $[0,1]$. https://doi.org/10.1093/logcom%2Fexaf043

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