ScalingStacks

Proof. By Corollary 2.64, every point in $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ has a neighbourhood system consisting of affinoid domains. Hence for any $z\in\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$, there exists an affinoid domain neighbourhood $V(z)$. By Lemma 4.1, $z$ has an open neighbourhood $\text{Int}^{\mathrm{top}}_{\mathfrak{M}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$, so we can assume that each $V(z)$ is contained in this open set.

One forms a covering by open sets

$$\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\subseteq\bigcup_{z\in\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))}\text{Int}^{\mathrm{top}}_{\mathfrak{M}}V(z).$$

Since the left hand side is a compact set by Proposition 2.21, there exist finitely many points $\{z_{1},\dots,z_{m}\}$ such that $\{\text{Int}^{\mathrm{top}}_{\mathfrak{M}}V(z_{i})\}_{i\in\{1,\dots,m\}}$ form a covering of $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$. Let $W_{\epsilon}$ be the union of affinoid domains $\{V(z_{i})\}_{i\in\{1,\dots,m\}}$, then it is a special domain, and satisfies the desired inclusion conditions. ∎