Parameterized stratification and piece number of D-semianalytic sets

We obtain results on the geometry of D-semianalytic and subanalytic subsets over a complete, non-trivially valued non-Archimedean field K, which is not necessarily algebraically closed. Among the results are a parameterized smooth stratification theorem and several results concerning the dimension of the D-semianalytic and subanalytic sets. We also extend Bartenwerfer's definition of piece number for analytic K-varieties to D-semianalytic sets and prove the existence of a uniform bound for the piece number of the fibers of a D-semianalytic set. We also establish a connection between the piece number and complexity of D-semianalytic sets which are subsets of the line and thereby give a simpler proof of the Complexity Theorem of Lipshitz and Robinson. We finish by proving that for each D-semianalytic X, there is a semialgebraic Y such that one dimensional fibers of X are among the one dimensional fibers of Y. This is an analogue of a theorem by van den Dries, Haskell and Macpherson.