Example 1.2.3.11. Let $n$ be a nonnegative integer. Combining Example 1.2.3.8 with Proposition 1.2.3.10, we see that the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^{n}$ induces a homeomorphism from $| \operatorname{\partial \Delta }^{n} |$ to the boundary of the topological $n$-simplex $| \Delta ^{n} |$, given by
\[ \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: t_ j = 0 \text{ for some $j$} \} . \]