Computably isometric spaces

Abstract

We say that an uncountable metric space is computably categorical if every two computable structures on this space are equivalent up to a computable isometry. We show that Cantor space, the Urysohn space, and every separable Hilbert space are computably categorical, but the space $ \mathcal{C}[0,1]$ of continuous functions on the unit interval with the supremum metric is not. We also characterize computably categorical subspaces of $\mathbb{R}^n$, and give a sufficient condition for a space to be computably categorical. Our interest is motivated by classical and recent results in computable (countable) model theory and computable analysis.