Let $X$ be a normal variety over $K$. We denote by $X^{\an}$ the Berkovich analytification of $X$. Set-theoretically, it consists of pairs $x=(\xi_{x},v_{x})$ where $\xi_{x}\in X$ and $v_{x}$ is a real-valued valuation on the residue field at $\xi_{x}$ extending the valuation $\ord_{t}$ on $K$. We denote by $\mathscr{H}(x)$ the completion of the residue field at $\xi_{x}$ with respect to $v_{x}$. We endow $X^{\an}$ with the coarsest topology such that

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  the forgetful map $\iota:X^{\an}\rightarrow X$, which maps $x=(\xi_{x},v_{x})$ to $\xi_{x}$, is continuous;

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  for any Zariski open $U\subseteq X$ and any function $f\in\mathcal{O}_{X}(U)$, the map:

  $$\lvert f\rvert:U^{\an}\coloneqq\iota^{-1}(U)\rightarrow\mathbb{R},$$

  which evaluates $f$ at $x$ associating the value $\lvert f\rvert(x)\coloneqq\exp(-v_{x}(f(\xi_{x})))$, is continuous.

This makes $X^{\an}$ a Hausdorff topological space, which is compact if and only if $X$ is proper over $K$.

Assume that $X/K$ is proper, and let $\mathscr{X}/R$ be a proper model of $X$. By the valuative criterion of properness, for any $x=(\xi_{x},v_{x})\in X^{\an}$ there is a unique lift of the point $\xi_{x}$ to the valuation ring $\mathscr{H}(x)^{\circ}$ of $\mathscr{H}(x)$:

The image of the closed point of $\Spec\mathscr{H}(x)^{\circ}$ under the extended morphism $\Spec\mathscr{H}(x)^{\circ}\rightarrow\mathscr{X}$ is called the center (or specialization) of $x$ and denoted by $c_{\mathscr{X}}(x)$. The map $c_{\mathscr{X}}:X^{\an}\longrightarrow\mathscr{X}_{k}$ turns out to be anticontinuous, i.e. the preimage of an open subset of $X^{\an}$ by $c_{\mathscr{X}}$ is closed in $\mathscr{X}_{k}$.