Non-archimedean tame topology and stably dominated types

Understand

Let $V$ be a quasi-projective algebraic variety over a non-archimedean valued field.

• We introduce topological methods into the model theory of valued fields, define an analogue $\hat {V}$ of the Berkovich analytification $V^{an}$ of $V$, and deduce several new results on Berkovich spaces from it.
• In particular we show that $V^{an}$ retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on $V$.
• When $V$ varies in an algebraic family, we show that the homotopy type of $V^{an}$ takes only a finite number of values.