> [!definition] > > Let $S = \bracs{1, \cdots, k}$ be a finite [[Set|set]]. A **simplicial complex** over $S$ is a family $\cc \subset 2^S$ such that for every $A \in \cc$ and $B \subset A$ with $B \ne \emptyset$, $B \in \cc$ as well. > [!definition] > > Let $(S, \cc)$ be a simplicial complex and $\seqf[k]{e_j} \subset \real^{k}$ be the standard [[Basis|basis]]. For each $n \in \nat$ and $A \in \cc$ with $|A| = n$, let > $ > D_A^{n - 1} = \text{Conv}(\bracs{e_j: j \in A}) > $ > and > $ > X^{n - 1} = \bigcup_{A \subset \cc, |A| = n - 1}D_A^{n - 1} > $ > with attaching maps $\iota|_{D_A^n \cap X^{n-2}}: \partial D_A^{n-1} \to X^{n - 2}$ if $n \ge 1$, then $\bracs{X^n: 0 \le n \le k - 1}$ forms a family of skeletons for a [[CW-Complex|CW-complex]] $X$, known as the **geometric realisation** of $(S, \cc)$.