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 $V^{\mathrm{an}}$ of any $d$ -dimensional quasi-projective scheme $V$ over $K$ embeds in ${\mathbb{R}}^{2d+1}$. If, moreover, the value group of $K$ is dense in ${\mathbb{R}}_{>0}$ and $V$ is a curve, then we describe the homeomorphism type of $V^{\mathrm{an}}$ by using the theory of local dendrites.