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

2025/08/08 by Badaev, Serikzhan, Bazhenov, Nikolay, Goncharov, Sergey +2
#03C57 (Secondary) #03D78 (Primary) 03D45 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2508.06187

Abstract

We prove that there exists a Σ⁰₁ 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].

Citations

• Strong computable type
• Computable topological abelian groups
• Degree spectra of homeomorphism types of compact Polish spaces
• Computing the exponent of a Lebesgue space

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