A topological ring $A$ is called adic if there is an ideal $\mathfrak{a}\subseteq A$ such that the ideals $(\mathfrak{a}^{n})_{n\in\mathbb{N}}$ form a neighbourhood basis for $0$. We call $\mathfrak{a}$ a defining ideal. Let $A$ be an adic, complete, separated ring with finitely generated defining ideal $\mathfrak{a}$. The affine formal scheme of $A$ is the locally topologically ringed space $\Spf(A)=(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ where $\mathfrak{X}$ and $\mathcal{O}_{\mathfrak{X}}$ are defined as follows: $\mathfrak{X}$ is the set of all open prime ideals of $A$. As a prime ideal is open if and only if it contains $\mathfrak{a}$, we may identify $\mathfrak{X}$ with $\Spec(A/\mathfrak{a})\subseteq\Spec(A)$ and we endow $\mathfrak{X}$ with the topology induced by the Zariski topology on $\Spec(A)$. Moreover we define

A formal scheme is a locally topologically ringed space $(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ such that for each $x\in\mathfrak{X}$ there is an open neighbourhood $\mathfrak{U}$ of $x$ with $\left(\mathfrak{U},\mathcal{O}_{\mathfrak{X}}\Big|_{\mathfrak{U}}\right)$ isomorphic to an affine formal scheme.

Now let $\mathfrak{a}$ be a defining ideal of $K^{\circ}$. A topological $K^{\circ}$-algebra $A$ is called admissible, if $\left\{a\in A\;\Big|\;\mathfrak{a}^{n}\cdot a=0\text{ for some }n\in\mathbb{N}\right\}=\{0\}$ i.e. $A$ does not have $K^{\circ}$-torsion and if $A$ is isomorphic to a $K^{\circ}$-algebra of the form $K^{\circ}\langle\zeta_{1},...,\zeta_{n}\rangle/(a_{1},...,a_{m})$ endowed with the $\mathfrak{a}$-adic topology. A formal $K^{\circ}$-scheme $\mathfrak{X}$ is called admissible if there is a locally finite open cover $(\mathfrak{U}_{i})_{i\in I}$ of $\mathfrak{X}$ with $\mathfrak{U}_{i}=\Spf(A_{i})$ for admissible $K^{\circ}$-algebras $A_{i}$.

Let $\mathfrak{X}=\Spf(A)$ be an admissible formal affine $K^{\circ}$-scheme. The analytic generic fibre of $\mathfrak{X}$ is defined as $\mathfrak{X}^{\textup{an}}:=\mathcal{M}(A\otimes_{K^{\circ}}K)$, where $\mathcal{M}(\cdot)$ denotes the Berkovich spectrum (cf. [Ber90, 1.2]). The special fibre of $\mathfrak{X}$ is given by $\tilde{\mathfrak{X}}:=\Spec(A\otimes_{K^{\circ}}k)$, where $k:=K^{\circ}/K^{\circ\circ}$ is the residue field of $K$. For an admissible formal $K^{\circ}$-scheme $\mathfrak{X}$ one obtains the generic and the special fibre by a gluing process. There is a canonical surjective reduction map $\red:\mathfrak{X}^{\textup{an}}\rightarrow\tilde{\mathfrak{X}}$, see [GRW17, §2.13].