respect to the product topology on $\{0,1\}^{\mathbb{N}_0}$, i.e. open sets are generated by cylinder sets. We have the following Lemma:

\[lem-naturalcomplex\] There is a continuous map $$\begin{aligned}
\Psi\colon{\mathcal{SC}_*}&\longrightarrow \{\text{subspaces of } \Delta({{\mathbb N}}_0) \}\subseteq\{0,1\}^{{{\mathbb N}}_0},\\
[K,x]&\longmapsto (K,x)_\mathbb{N}\end{aligned}$$ such that $[(K,x)_{{\mathbb N}},0]=[K,x]$.

Let $[K,x]\in{\mathcal{SC}_*}$. That there is a subcomplex $\Lambda$ of ${\Delta(\mathbb{N}_0)}$ with $(\Lambda, 0)\in[K,x]$ is obvious, we obtain such a subcomplex by enumerating the vertices of $K$ in an arbitrary way and label the root with $0$. Thus, we restrict to this kind of subcomplexes. Given two such subcomplexes $\Lambda,\Xi\in\{0,1\}^{{{\mathbb N}}_0}$. Then $\Lambda<\Xi$ if, considered as sequences, $\Lambda<_{\mathrm{lex}}\Xi$ with respect to the lexicographic order, i.e. we look at the first position where the sequences differ, if $\Lambda$ has a 1 there, we say $\Lambda<\Xi$. $\Psi$ maps $[K,x]$ to the smallest complex $(K,x)_\mathbb{N}\in\{0,1\}^{{{\mathbb N}}_0}$ with respect to this order.\
**Claim:** There exists a unique minimal subcomplex $(K,x)_{{\mathbb N}}\subset {\Delta(\mathbb{N}_0)}$ for every $[K,x]\in{\mathcal{SC}_*}$.\
For a finite simplicial complex it follows from the well-ordering principle that a unique minimal subcomplex of ${\Delta(\mathbb{N}_0)}$ exist. Let $[K,x]$ by the rooted isomorphism class of an infinite simplicial complex $K$ and consider the $r$-ball $B_{(K,x)}(r)$ around $x$ in $K$. We claim that $\Psi(B_{(K,x)}(r))=B_{\Psi(B_{(K,x)}(r+1))}(r)$ and hence the minimal 