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

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 $[0,1]$.