Metric Geometry in a Tame Setting

Abstract

The thesis is about the topology and geometry of metric spaces definable in an o-minimal expansion R of an ordered field (R, <, +, •). A definable metric space is a pair (X, d) consisting of a definable set X ⊆ R k and a definable (R, +, <)-valued metric. If X ⊆ R k is definable and e is the restriction of the usual euclidean metric on R k to X then (X, e) is a definable metric space, in this way the geometry of definable sets may be considered as a special case of the geometry of definable metric spaces. Examples of definable metric spaces whose geometry is unlike that of any definable set are given by the hyperbolic plane (Rexpdefinable) and certain subriemannian spaces (Ran-definable). The main theorem of the thesis is the following: Let (X, d) be a definable metric space.