$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition Let $S_{\bullet }$ be a simplicial set. Then $S_{\bullet }$ is the disjoint union of its connected components.

Proof. Let $\sigma $ be an $n$-simplex of $S_{\bullet }$; we wish to show that there is a unique connected component of $S_{\bullet }$ which contains $\sigma $. This follows from Proposition, applied to the map $\Delta ^ n \rightarrow S_{\bullet }$ classified by $\sigma $ (since the standard $n$-simplex $\Delta ^ n$ is connected; see Example $\square$