ScalingStacks

Definition 2.19. Let $\mathcal{A}$ be a $k$-Banach algebra. The Berkovich spectrum $\mathfrak{M}(\mathcal{A})$ is the following topological space: the points, denoted by $z$, are bounded multiplicative $k$-algebra seminorms $\vvvert\mathord{\cdot}\vvvert_{z}$ on $\mathcal{A}$, and the topology is the weakest topology on this set of points, for which all $\mathbb{R}_{\geq 0}$-valued functions of the form $z\mapsto\vvvert f\vvvert_{z}$ are continuous for any $f\in A$. This topology is called the canonical topology. For any subset $V$ of $\mathfrak{M}(\mathcal{A})$, we denote by $\text{Int}^{\mathrm{top}}(V)$ the topological interior of $V$. This topological interior is to be compared with the notion of interior of an affinoid subdomain in an affinoid domain (see [Ber, Definition 2.5.7]), which we do not use in this article.