closed ball is the center of this ball and, as a consequence, that the intersection of two balls is empty or one ball is included in the other. It will play a major role in the understanding of tree structure in the Berkovich space.

To define precisely what is this Berkovich space, we would have to talk about valuations, semi norms but one would want first to forget about the formalism and concentrate on the keys for understanding this space. Let’s denote by $\P^1_\LL\approx\LL\cup\{\infty\}$ the projective space over $\LL$.

$ \P^1 _{Berk}$ is a compactification of $\P^1_\LL$ called the Berkovich projective line.

Here is the important remark due to Berkovich:

\[prethm\] Almost every element of $ \P^1 _{Berk}$ is identified to a closed ball $$B(z_0,r)=\{z\in \P^1_\LL ~|~ |z-z_0|\leq r \}.$$

From this we can deduce that $\P^1 _{Berk}$ has a real tree structure. A way to represent the space $\P^1 _{Berk}$ is to consider it as the quotient of the product $" {\LL} \times (\R^+\cup\{\infty\} )"$ (Center$\times$ Radius) by the relation provided in Lemma \[ball\]. The real structure of this space comes from the vertical line that all behave well to the quotient.

For example take $a\neq b$ in $\LL$. 