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  "body": "\n## Statement\n\nLet $X$ be a [[paracompact space|ZH29VRRRJ6]] and let $\\mathcal{U} = (U_i)_{i \\in I}$ be an [[open cover|049PXYHW1T]] of $X$.\nAssume that every nonempty intersection $U_{i_0} \\cap \\cdots \\cap U_{i_k}$ of finitely many members of $\\mathcal{U}$ is [[contractible|R5WP8Z69Q8]].\nThen $X$ is [[homotopy equivalent|XZG6MTSWCN]] to the geometric realization of the [[nerve|4P3S0DRZXB]] $N(\\mathcal{U})$.\n\n## Remarks\n\nThe proof uses a partition of unity subordinate to $\\mathcal{U}$ to define a map $X \\to \\abs{N(\\mathcal{U})}$, sending $x$ to the point with barycentric coordinates $(\\varphi_i(x))_{i \\in I}$.\nOne then shows that this map is a homotopy equivalence.\nA standard way compares both spaces with a common space built from the intersections, whose projections to $X$ and to $\\abs{N(\\mathcal{U})}$ have contractible fibres.\n\nExample: cover $S^1$ by three open arcs, consecutive arcs overlapping in an arc, and no point in all three.\nEvery nonempty intersection is an arc, which is contractible, and the nerve is the boundary of a triangle, a circle, as the theorem predicts.\n\nThe contractibility hypothesis is necessary.\nCover $S^1$ by two open arcs, each a bit larger than a half circle.\nTheir intersection is two disjoint arcs, which is not contractible, and the nerve is a single edge, which is contractible, while $S^1$ is not.\n\nVersions of the theorem hold for finite covers of a simplicial complex by subcomplexes whose nonempty intersections are contractible.\n\n## Explanations\n\n### L1 human\n\nIf a space is covered by simple pieces whose overlaps are also simple, then the overlap pattern alone determines the shape of the space up to homotopy.\nThe overlap pattern is recorded by the nerve, a combinatorial object.\nSo one can compute topological invariants of the space from a finite list of which pieces meet.\n\n### L2 human\n\nA cover with contractible finite intersections is called a good cover.\nEvery smooth manifold has a good cover by geodesically convex balls for a Riemannian metric.\nSo the nerve theorem shows that a compact smooth manifold has the homotopy type of a finite simplicial complex.\nIn topological data analysis, the Čech complex of a point cloud is the nerve of a cover by balls.\nThe nerve theorem shows that it recovers the homotopy type of the union of the balls.\n",
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