Berkovich spaces embed in Euclidean spaces

Abstract

Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V[superscript an] of any d-dimensional quasi-projective scheme V over K embeds in R[superscrip 2d+1]. If, moreover, the value group of K is dense in R>0 and V is a curve, then we describe the homeomorphism type of V[superscript an] by using the theory of local dendrites.