Effective compactness and orbits of points under the isometry group

Abstract

We examine conditions under which effective separating sequences in a metric space are equivalent or equivalent up to an isometry. We prove that two effective separating sequences in an effectively compact metric space are equivalent in the case when there is a set which is computable with respect to both sequences and has the property that there are only finitely many isometries of the ambient space which preserve it. Furthermore, we prove that in an effectively compact computable metric space the orbit of a computable point under the isometry group is a co-c.e. set. In general, a compact subspace of an Euclidean space need not be computably categorical. Using the result for orbits we prove that any effectively compact subspace of R2 and R3 is computably categorical.