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The boundary subcomplex $\partial {\Delta }^n$ consists of all proper faces of the $n$-simplex. Its geometric realization of a simplicial complex is homeomorphic to the sphere $S^{n-1}$, while $|{\Delta }^n|$ is homeomorphic to the closed disk $D^n$.

Let $K_n$ be the abstract simplex on the vertex setthus every subset of these vertices is a simplex. Let $L_{n-1}$ be its boundary subcomplex of a simplex, consisting of all proper faces, so the only simplex of $K_n$ absent from $L_{n-1}$ is the full $n$-simplex $[v_0\cdots v_n]$.

The geometric realization of a simplicial complex $|K_n|$ is the usual solid geometric $n$-simplex, which is homeomorphic to the closed disk $D^n$. Under such a homeomorphism its topological boundary is $|L_{n-1}|$, and

The geometric realization of a simplicial complex consists of the filled triangle $⟨v_1,v_2,v_3⟩$ together with the two extra edges $⟨v_1,v_4⟩$ and $⟨v_2,v_4⟩$. Thus the triangle on $v_1,v_2,v_4$ has only its boundary: