Effective Dispersion in Computable Metric Spaces

Abstract

We investigate the relationship between computable metric spaces $(X,d,\alpha )$ and $(X,d,\beta ),$ where $(X,d)$ is a given metric space. In the case of Euclidean space, $\alpha $ and $\beta $ are equivalent up to isometry, which does not hold in general. We introduce the notion of effectively dispersed metric space. This notion is essential in the proof of the main result of this paper: $(X,d,\alpha )$ is effectively totally bounded if and only if $(X,d,\beta )$ is effectively totally bounded, i.e. the property that a computable metric space is effectively totally bounded (and in particular effectively compact) depends only on the underlying metric space.