the residue field $k(p_x)$ at the point $p_x$ extending the absolute value of $K$. The topology on ${X^{\rm an}}$ is the weakest topology making the canonical map ${X^{\rm an}}\to X$ continuous and, for every open set $U$ of $X$ and section $f \in \mathcal{O}_X(U)^\times$, the map $U^{\operatorname{an}} \to {\mathbf}R$ given by $$(p_x, |\cdot|_x) \mapsto |f(x)| \coloneqq |f(p_x)|_x$$ is continuous.

### Classification of points

When $X$ is a curve, the points of ${X^{\rm an}}$ can be classified into four types.

If $p_x$ is a closed point of $X$, then $k(p_x) = K$ and $|\cdot|_x = |\cdot|_K$ is the only absolute value we can take. In this way, we view $X(K)$ as a canonical subset of ${X^{\rm an}}$. Points in $X(K)$ are called type I points of ${X^{\rm an}}$.

If $p_x$ is the generic point of $X$ and ${\mathscr}{H}(x)$ is the completion of $k(p_x)$ with respect to $|\cdot|_x$. Then we say $(p_x, |\cdot|_x)$ is a type II point if $\operatorname{trdeg}({\widetilde}{{\mathscr}H}(x)/{\widetilde}{K}) = 1$ where ${\widetilde}{\cdot}$ denotes the residue field.

The terminology of type I and type II points is due to Thuiller [@Thu] following Berkovich’s original classification [@Berk]. There is also a notion of type III and IV points (*loc. cit.*) which we 