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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $\\mathcal{U} = (U_i)_{i \\in I}$ be a family of subsets of $X$, usually an [[open cover|049PXYHW1T]].\nThe *nerve* $N(\\mathcal{U})$ is the abstract [[simplicial complex|V8GZ0W1ZZ5]] with vertex set $\\set{i \\in I : U_i \\ne \\emptyset}$ whose faces are the finite nonempty subsets $J \\subseteq I$ such that\n$$\\bigcap_{j \\in J} U_j \\ne \\emptyset.$$\n\n## Remarks\n\nIf $J$ is a face and $J' \\subseteq J$ is nonempty, then $\\bigcap_{j \\in J'} U_j \\supseteq \\bigcap_{j \\in J} U_j \\ne \\emptyset$, so $J'$ is a face; this is why $N(\\mathcal{U})$ is a simplicial complex.\n\nExample: cover the circle $S^1$ by three open arcs $U_0$, $U_1$, $U_2$, each a bit longer than one third of the circle.\nChoose them so that consecutive arcs overlap and no point lies in all three.\nEach pair $\\set{i, j}$ is a face, and $\\set{0, 1, 2}$ is not, so $N(\\mathcal{U})$ is the boundary of a triangle.\n\nNon-example: the complex with vertices $0, 1, 2$ and all subsets as faces is not the nerve of this cover, since it contains $\\set{0, 1, 2}$ although $U_0 \\cap U_1 \\cap U_2 = \\emptyset$.\n\nThe same definition applies to any family of sets; the nerve of a closed cover or of a family of convex sets is often used in combinatorics.\n",
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