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  "body": "\n## Definition\n\nLet $V$ be a [[finite set|070R982G4B]].\nAn *(abstract) simplicial complex* on the vertex set $V$ is a family $\\Delta$ of subsets of $V$ such that:\n\n1. If $\\sigma \\in \\Delta$ and $\\tau \\subseteq \\sigma$, then $\\tau \\in \\Delta$.\n2. $\\set{v} \\in \\Delta$ for every $v \\in V$.\n\nThe elements of $\\Delta$ are its *faces*; a face $\\sigma$ has dimension $\\abs{\\sigma} - 1$, and the dimension of $\\Delta$ is the largest dimension of a face.\nThe maximal faces are called *facets*.\n\n## Remarks\n\nExample: on $V = \\set{1, 2, 3}$, let $\\Delta$ be the family of all subsets of $V$ except $V$ itself.\nA subset of a set with at most $2$ elements has at most $2$ elements, so clause 1 holds.\nClause 2 holds, and $\\Delta$ is the boundary of a triangle, of dimension $1$.\n\nNon-example: on $V = \\set{1, 2, 3}$, the family $\\set{\\emptyset, \\set{1}, \\set{2}, \\set{3}, \\set{1, 2}, \\set{1, 2, 3}}$ fails clause 1, since $\\set{1, 3} \\subseteq \\set{1, 2, 3}$ is missing.\n\nA nonempty complex contains the empty face, of dimension $-1$.\nSome authors allow an infinite vertex set, keeping every face finite.\nA geometric realization places the vertices as affinely independent points in some $\\R^N$ and replaces each face by the convex hull of its points.\n",
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