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I'm reading Berkovich's book on analytic spaces. The notion of relative interior confuses me. Is there anyway to see how it "looks like"? For instance, if $r <1$, what is the relative interior of \begin{equation} M ( \mathbb{C}_p \{r^{-1} T \} ) \to M ( \mathbb{C}_p \{ T \} ) \end{equation} and how one should view it on the pictorial description of $M ( \mathbb{C}_p \{ T \} )$, i.e. the famous picture which looks like an infinite tree.

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Let me begin with the absolute interior. Let me fix first a complete non-archimedean field $k$ (take $k=\mathbb{C}_p$ if you like). For a closed disk $D^+(0;r_1,\dots,r_n)$ (center 0, polyradius $(r_1,\dots,r_n)$), I will call ``naive interior'' the open disk $D^-(0;r_1,\dots,r_n)$.

Now, let $Y=\mathcal{M}(B)$ be a $k$-affinoid space. By definition, you may realize it as a Zariski-closed subset of some closed disk $D^+$. The first idea one could have is to define the interior of $Y$ as the set of points that belong to the naive interior of $D^+$. Actually, this depends on the chosen presentation. So you are led to define the absolute interior Int$(Y/k)$ of $Y$ as the set of points that belong to the naive interior of a closed disk $D^+$ for some presentation of $Y$ as as Zariski-closed subset of $D^+$.

Let us consider a simple example where $k=\mathbb{C}_p$ and $B =\mathbb{C}_p\{T\}$. Topologically, this disk is a tree. The root (i.e. the Gauss point), I will denote $\eta_1$. If you write $Y=D^+(0;1)$, you see that $D^-(0;1)$ belong to the interior of $Y$. This is an open branch out of the point $\eta_1$. Changing coordinate gives you another presentation and another open disk $D^-(a;1)$ (i.e. another open branch) in the interior. This way, you show that $Y\setminus\{\eta_1\}$ belongs to the interior of $Y$. It is actually equal to it.

As regards the relative interior, you just have to replace $k$ by a $k$-affinoid algebra $A$, $B$ by an $A$-affinoid algebra and disks by relative disks over $A$. In your situation, by the same kind of arguments, you will find that the relative interior is the complement of the point $\eta_r$. Actually, Berkovich shows more generally that if $Y \to X$ is the inclusion of an affinoid domain, then Int$(Y/X)$ coincides with the topological interior of $Y$in $X$.

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