Effective subsets under homeomorphisms of \(\mathbb{R}^n\) (Q897661)

This paper studies the (non-)preservation of computability of a compact subset of \(\mathbb{R}^n\) under arbitrary (i.e. not necessarily computable) homeomorphisms. Strengthening the result that any non-empty computable compact subset of \(\mathbb{R}^n\) is homeomorphic to a non-computable compact subset of \(\mathbb{R}^n\), the paper proves a conjecture of Braverman. Namely, there is a computably (co-)enumerable compact subset K of \([0,1]^n\) such that under no homeomorphism \(f\) on \(\mathbb{R}^n\) do we have that \(f(K)\) is computable.