Proof: [036H]

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Proof: It is enough to show that for every $x\in V$ , there is a non-negative smooth function $\phi$ with compact support in $V$ and with $\phi(x)>0$ . Since ${X^{\rm an}}$ is a locally compact Hausdorff space which is also $\sigma$ -compact, the open subset $V$ is paracompact and hence standard arguments from differential geometry yield the existence of the desired partition of unity (see [Wa83], Theorem 1.11).

To prove the crucial claim at the beginning of the proof, we may assume that $V$ is coming from a tropical chart $(V,\varphi_{U})$ (see Proposition 4.16). Then $\Omega:={\rm trop}_{U}(V)$ is a open subset of ${\rm Trop}(U)$ with ${\rm trop}_{U}^{-1}(\Omega)=V$ and hence there is an open subset $\tilde{\Omega}$ in $(N_{U})_{\mathbb{R}}$ with $\Omega=\tilde{\Omega}\cap{\rm Trop}(U)$ . There is a smooth non-negative function $f$ on $(N_{U})_{\mathbb{R}}$ with compact support in $\tilde{\Omega}$ such that $f({\rm trop}_{U}(x))>0$ . Since the tropicalization map is proper, the smooth function $\phi:=f\circ{\rm trop}_{U}$ has compact support in $V$ and hence $\phi$ fulfills the claim. $\square$