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Remark Let $I$ be a finite set and let $\operatorname{\raise {0.1ex}{\square }}^{I}$ be the $I$-cube of Notation Then the geometric realization $| \operatorname{\raise {0.1ex}{\square }}^{I} |$ can be identified with the topological space $\prod _{i \in I} [0,1]$. In particular, the geometric realization $| \operatorname{\raise {0.1ex}{\square }}^{n} |$ is homeomorphic to the standard cube

\[ \{ (t_1, t_2, \ldots , t_ n ) \in \operatorname{\mathbf{R}}^{n}: 0 \leq t_ i \leq 1 \} . \]

This is a tautology in the case $n = 1$, and follows in general from the compatibility of geometric realizations with products of finite simplicial sets (see Corollary