A non-computable c.e. closed subset of $[0,1]$
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916887724556288 |
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| author | Badaev, Serikzhan Bazhenov, Nikolay Goncharov, Sergey Kalmurzayev, Birzhan Melnikov, Alexander |
| author_facet | Badaev, Serikzhan Bazhenov, Nikolay Goncharov, Sergey Kalmurzayev, Birzhan Melnikov, Alexander |
| contents | We prove that there exists a $Σ^0_1$ 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 $[0,1]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A non-computable c.e. closed subset of $[0,1]$ Badaev, Serikzhan Bazhenov, Nikolay Goncharov, Sergey Kalmurzayev, Birzhan Melnikov, Alexander Logic 03D78 (Primary) 03D45, 03C57 (Secondary) We prove that there exists a $Σ^0_1$ 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 $[0,1]$. |
| title | A non-computable c.e. closed subset of $[0,1]$ |
| topic | Logic 03D78 (Primary) 03D45, 03C57 (Secondary) |
| url | https://arxiv.org/abs/2508.06187 |