Overconvergent subanalytic subsets in the framework of Berkovich spaces

We study the class of overconvergent subanalytic subsets of a k-affinoid space X when k is a non-archimedean field. These are the images along the projection X x B-n -> X of subsets defined by inequalities between functions on X x B-n which are overconvergent in the variables of B-n. In particular, we study the local nature, with respect to X, of overconvergent subanalytic sub-sets. We show that they behave well with respect to the Berkovich topology, but not the G-topology. This gives counterexamples to previous results on the subject, and a way to correct them. Moreover, we study the case dim. (X) = 2, for which a simpler characterisation of overconvergent subanalytic subsets is proven