To any scheme $X$ of finite type over a Banach ring $A$, Berkovich associates an analytification⁹⁹ 9 We use the term analytification even though we shall only consider $X^{\mathrm{An}}$ as a topological space. In particular, $X^{\mathrm{An}}$ only depends on the reduced scheme structure of $X$. $X^{\mathrm{An}}$, a locally compact topological space with a continuous morphism $X^{\mathrm{An}}\to{\mathcal{M}}(A)$, defined as follows.

When $X=\operatorname{Spec}B$ is affine, with $B$ a finitely generated $A$-algebra, $X^{\mathrm{An}}$ is defined as the set of multiplicative seminorms $|\cdot|_{x}$ on $B$ whose restriction to $A$ is bounded by the given norm on $A$, i.e. belongs to ${\mathcal{M}}(A)$. The topology on $X^{\mathrm{An}}$ is the weakest one for which $x\mapsto|f|_{x}=|f(x)|$ is continuous for every $f\in B$.

In the general case, the analytification $X^{\mathrm{An}}$ is defined by gluing together the analytifications of an affine open cover, and yields a covariant functor $X\mapsto X^{\mathrm{An}}$. If $X\hookrightarrow Y$ is an open (resp. closed) embedding, then so is $X^{\mathrm{An}}\hookrightarrow Y^{\mathrm{An}}$. If $X\to Y$ is surjective, then so is $X^{\mathrm{An}}\to Y^{\mathrm{An}}$.

The topological space $X^{\mathrm{An}}$ is Hausdorff (resp. compact) if $X$ is separated (resp. projective). The assignment $x\mapsto{\mathfrak{p}}_{x}$ above globalizes to a continuous map

$$\pi\colon X^{\mathrm{An}}\to X,$$

where $X$ is equipped with the Zariski topology.

When $A$ is a valued field, it is more common to write $X^{\mathrm{an}}$ instead of $X^{\mathrm{An}}$ [Berk90].