On metric types that are definable in an o-minimal structure. to appear in Journal of Symbolic Logic

Abstract. In this paper we study the metric spaces that are definable in a polynomially bounded o-minimal structure. We prove that the family of metric spaces definable in a given polynomially bounded o-minimal structure is characterized by the valuation field Λ of the structure. In the last section we prove that the cardinality of this family is that of Λ. In particular these two results answer a conjecture given in [SS] about the countability of the metric types of analytic germs. The proof is a mixture of geometry and model theory. §0. Introduction. Given a subset of Rn, definable in an o-minimal structure, we may consider it as a metric space if we endow it with the induced metric of Rn. The classification of such subspaces up to bi-Lipschitz homeomorphisms goes back to T. Mostowski’s work on complex analytic sets [M]. He was motivated to find a notion of stability more precise than topological stability