On the Topology of Metric Spaces Definable in o-minimal expansions of fields

We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably homeomorphic to a definable set equipped with its euclidean topology. We show that almost every point in a definable metric space has a neighborhood which is definably homeomorphic to an open definable subset of euclidean space. We also show that over an expansion of the real field, the completion of a definable metric space is definable and characterize the topological dimension of a definable metric space.