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

TL;DR

This paper constructs a specific computably enumerable closed subset of the interval [0,1] that cannot be homeomorphic to any computably compact space, revealing limitations in the computable structure of certain topological spaces.

Contribution

It demonstrates the existence of a non-computably homeomorphic c.e. closed subset of [0,1] and shows the non-arithmetical nature of the index set of c.e. subspaces with computably compact presentations.

Findings

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

The index set of c.e. subspaces with computably compact presentations is not arithmetical

Results extend to general computable Polish spaces

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 $[0,1]$.