Note that we have a natural section ${\mathbb{R}}^{r}\rightarrow{T^{\rm an}}$ of the tropicalization map. It is given by mapping the point $\omega\in{\mathbb{R}}^{r}$ to the weighted Gauss norm $\eta_{\mathbf{s}}$ associated to ${\mathbf{s}}:=(e^{-\omega_{1}},\dots,e^{-\omega_{r}})$. It follows from [Be90], Example 5.2.12, that this section is a homeomorphism of ${\mathbb{R}}^{r}$ onto a closed subset of ${T^{\rm an}}$ which is the skeleton $S({T^{\rm an}})$ introduced in 4.3. In this way, we may view the tropicalization map as a map from ${T^{\rm an}}$ onto $S({X^{\rm an}})$. Then it is shown in [Be90], §6.3, that the tropicalization map is a strong deformation retraction of ${T^{\rm an}}$ onto the skeleton $S({T^{\rm an}})$. This point of view is used very rarely in our paper.