Remark 1.2.4.6. Let $0 \leq i \leq n$ be integers with $n > 0$. It follows from Proposition 1.2.3.10 that the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ induces a homeomorphism from the geometric realization $| \Lambda ^{n}_{i} |$ to the closed subset of $| \Delta ^{n} |$ given by
\[ \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: t_ j = 0 \text{ for some $j \neq i$} \} . \]