Geometry of D-semianalytic and subanalytic sets over complete non -Archimedean fields

This is an investigation of basic geometric properties of D-semianalytic and subanalytic sets over an arbitrary (i.e. not necessarily algebraically closed) non-trivially valued complete non-Archimedean fields, mostly in the characteristic 0 case. Main results include a Parameterized Normalization Theorem and a Parameterized Smooth Stratification Theorem for D-semianalytic sets as well as a Bounded Piece Number Theorem for fibers of a D-semianalytic set. Several properties that a well behaved dimension function is expected to satisfy are also proved to hold for a natural notion of dimension when applied to D-semianalytic and subanalytic sets. As an application of the Bounded Piece Number Theorem, a new proof of the Complexity Theorem of Lipshitz and Robinson is given at the end