ScalingStacks

Proof. By Proposition 2.94, the morphism $p(\boldsymbol{0})$ of schemes of finite type over $\spec k$ induces a continuous map betweeen the topological space of their analytification:

$$p(\boldsymbol{0})^{\mathrm{an}}:Tot(L^{\vee})^{\mathrm{an}}\rightarrow(\spec V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}}.$$

Since

$$p(\boldsymbol{0}):Tot(L^{\vee})\setminus\mathbb{O}\rightarrow\spec(V_{{\scriptscriptstyle\bullet}}(L))\setminus\boldsymbol{0}$$

is an isomorphism of schemes of finite type, its analytification induces a homeomorphism by Proposition 2.94

$$p(\boldsymbol{0})^{\mathrm{an}}:Tot(L^{\vee})^{\mathrm{an}}\setminus\mathbb{O}^{\mathrm{an}}\rightarrow(\spec V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}}\setminus\boldsymbol{0}^{\mathrm{an}}.$$

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