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

Erik Walsberg

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arXiv · 1510.07291

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

Erik Walsberg

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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.

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Erik Walsberg. 2015-11-11. On the Topology of Metric Spaces definable in o-minimal expansions of fields. https://arxiv.org/abs/1510.07291

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