A non-computable c.e. closed subset of [0,1]
S. A. Badaev, Nikolay A. Bazhenov, Sergey Savostyanovich Goncharov, Birzhan Kalmurzayev, Alexander Melnikov · Journal of Logic and Computation · 2025
Abstract We prove that there exists a $\varSigma ^{0}_{1}$ 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 $[0,1]$.