Example 3.2.2.3. For $n \geq 0$, the geometric realization $| \Delta ^ n / \operatorname{\partial \Delta }^ n |$ can be obtained from the topological $n$-simplex $| \Delta ^ n |$ by collapsing the boundary $| \operatorname{\partial \Delta }^ n |$ to a point (or by adding a new base point, in the degenerate case $n = 0$). It follows that $| \Delta ^ n / \operatorname{\partial \Delta }^ n |$ is homeomorphic to a sphere of dimension $n$.

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