said property. When $\Delta$ is thick and has no factor of rank $1$, one can actually show that the topology on $\operatorname{\mathrm{Vert}}\Delta$ is metrizable (see [@Grundhofer]\*[Proposition 6.14]{}). We will therefore always suppose that the topology comes from a metric. The **topological automorphism group** $\operatorname{\mathrm{Auttop}}(\Delta)$ of $\Delta$ is the group of all automorphisms of $\Delta$ whose restriction to each $\Delta_r$ are homeomorphisms. The group $\operatorname{\mathrm{Auttop}}(\Delta)$ is endowed with the topology induced from the compact-open topology of $C(\operatorname{\mathrm{Cham}}\Delta,\operatorname{\mathrm{Cham}}\Delta)$ (where $C(X,Y)$ denotes the set of continuous maps from $X$ to $Y$). Since $\operatorname{\mathrm{Cham}}\Delta$ is compact and metrizable, it also actually is the topology of uniform convergence.

We can now give the first properties of compact buildings. The following results all come from article [@Burns] of Burns and Spatzier and are almost direct consequences of the definition of compact buildings, except Lemma \[lemma:non-isolated\] whose proof is more technical.

\[proposition:opposite\] Let $\Delta$ be a compact spherical building of rank $k$. The set $$\{(R,R') \in \Delta_r^2 \mid R \text{ is opposite } R'\}$$ is open in $\Delta_r^2$ for each $1 \leq r \leq k$.

See [@Burns]\*[Proposition 1.9]{}.

\[proposition:projections\] Let $C$ be a chamber of a compact spherical building $\Delta$ and $R$ be a face of a 