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

Badaev S. Bazhenov N. Goncharov S. Kalmurzayev B. Melnikov A.
1 September 2025 Oxford University Press

Journal of Logic and Computation
2025 #35 Issue 6

We prove that there exists a Σ₁⁰ closed subset of [0, 1] which 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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Kazakh-British Technical University, School of Applied Mathematics, 59 Tole Bi Street, Almaty, 050000, Kazakhstan
Sobolev Institute of Mathematics, Laboratory of Computability Theory and Applied Logic, 4 Academician Koptyug Avenue, Novosibirsk, 630090, Russian Federation
Nazarbayev University, Department of Mathematics, 53 Qabanbaybatyr Avenue, Astana, 010000, Kazakhstan
Victoria University of Wellington, School of Mathematics and Statistics, Kelburn Parade, Wellington, 6012, New Zealand

Kazakh-British Technical University
Sobolev Institute of Mathematics
Nazarbayev University
Victoria University of Wellington

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