Computable type and computably categorical spaces

Abstract

We examine effective separating sequences on a metric space and, in particular, conditions under which on a metric space every two such sequences are equivalent up to an isometry. Such a metric space is called computably categorical. We prove that an effectively compact metric space $(X,d)$ is computably categorical if the space $\mathrm{Iso}(X,d)$ of all isometries of $(X,d)$ has computable type (which in particular holds if $\mathrm{Iso}(X,d)$ is a manifold). Using this, we prove that each effectively compact subspace of Euclidean space is computably categorical.