On semianalytic subsets of rigid analytic varieties

Abstract

We show some topological properties of semianalytic subsets of rigid analytic varieties: curve selection lemma, the closure $$\bar S$$ of a semianalytic subsetS is semianalytic, $$f(\bar S) = \overline {f(S)}$$ for every quasi-compact morphismf. As an application we show that a morphismf: X → Y of rigid analytic varieties is open at a pointx eX if and only if SpecO X,x → SpecO Y,f(x) is surjective.