Berkovich spaces embed in Euclidean spaces
  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^an of any d-dimensional quasi-projective scheme V over K embeds in R^{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^an by using the theory of local dendrites. 
Ehud Hrushovski, Fran\c{c}ois Loeser, Bjorn Poonen
math.AG math.GN math.NT