The *completed extended skeleton associated to ${\varphi}$* is the set $\Sigma({\varphi})$ of points in ${X^{\rm an}}$ that do not have an open neighborhood contained in $(X^\circ)^{\operatorname{an}}$ and isomorphic to an open disc. We write $\mathring{\Sigma}({\varphi})$ for the skeleton $\Sigma({\varphi})$ with its type I points removed.

If $Y$ is a product of $\mathbf{P}^1$’s, then ${\varphi}$ is defined by a set of rational functions and $X^\circ$ is the set of points that are neither zeroes nor poles of those functions. The skeleton $\Sigma({\varphi})$ contains all of those zeroes and poles as type I points.

Tropicalization of analytic curves
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If $Y$ is a projective space (or product of projective spaces) over $K$, then the map $\operatorname{Log}\colon Y \to \operatorname{Trop}(Y)$ defined in Section \[sec:classical-tropicalization\] extends to the analytification, $Y^{\operatorname{an}}$. We call this map $\operatorname{trop}\colon Y^{\operatorname{an}} \to \operatorname{Trop}(Y)$.

More generally, if $Y$ is a toric variety, then there is a map $\operatorname{trop}\colon Y^{\operatorname{an}} \to \operatorname{Trop}(Y)$. See [@Payne Section 3] for the definition.

When $Y = {\mathbf}P^1 = \operatorname{Proj}K[z_0, z_1]$, the map $\operatorname{trop}\colon {\mathbf}P^{1} \to {\mathbf}{TP}^1$ is given by $$\operatorname{trop}((p, |\cdot|_x)) = \log|z_1(p)|_x. \qedhere$$

When there is a closed embedding ${\varphi}$ of $X$ into the toric variety $Y$ (e.g. if $X$ is projective), we can 