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

Abstract

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.

Cited by in corpus (4)

• Directed sets and topological spaces definable in o-minimal structures
• Types, transversals and definable compactness in o-minimal structures
• Affiness of topological space definable in a definably complete uniformly locally o-minimal structure of the second kind
• Definable one dimensional topologies in o-minimal structures