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  "body": "\n## Statement\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]], let $E$ be an [[analytic subset|VMDTGTMN9P]] of $X$, and let $k \\ge 1$ be an integer.\nLet $A$ be an analytic subset of $X \\setminus E$ of [[pure dimension|C2R5Q2Q717]] $k$.\nSuppose that $\\dim E \u003c k$.\nThen the closure $\\overline{A}$ of $A$ in $X$ is an analytic subset of $X$, of pure dimension $k$.\n\n## Remarks\n\nThe question is local, so one may assume that $X$ is a neighbourhood of a point of $E$ in $\\C^n$.\nThe proof is by induction on the dimension of $E$ and uses projections.\nAfter a generic linear change of coordinates, the projection of $\\overline{A}$ to $\\C^k$ is proper near the point.\nThen $\\overline{A}$ is described by the canonical defining polynomials of this branched covering.\nTheir coefficients are bounded holomorphic functions outside a thin set, so they extend by the Riemann extension theorem.\nExample: $A = \\set{z_2 = 0} \\setminus \\set{0}$ is an analytic subset of $\\C^2 \\setminus \\set{0}$ of pure dimension $1$, and $E = \\set{0}$ has dimension $0$, so $\\overline{A}$, the $z_1$-axis, is analytic.\nThe hypothesis $\\dim E \u003c k$ is necessary.\nLet $E = \\set{z_1 = 0}$ in $\\C^2$ and $A = \\set{(z_1, e^{1/z_1}) : z_1 \\ne 0}$, an analytic subset of $\\C^2 \\setminus E$ of pure dimension $1 = \\dim E$.\nSince $e^{1/z_1}$ takes every nonzero value, and tends to $0$ along the negative real axis, in every neighbourhood of $z_1 = 0$, the closure $\\overline{A}$ is $A \\cup E$.\nIt is not analytic: the components of an analytic curve other than $E$ would meet $E$ in a discrete set, but $A$ accumulates at every point of $E$.\n\n## Explanations\n\n### L1 human\n\nSuppose an analytic set is known away from a smaller exceptional set.\nThe theorem says that if the exceptional set is of strictly smaller dimension, the analytic set extends across it: just add its limit points.\nFor instance, a curve defined outside a single point extends to a curve through that point.\n\n### L2 human\n\nThe theorem is a basic extension result in complex analytic geometry, in the spirit of the Riemann extension theorem for functions.\nIt gives a short proof of Chow's theorem, by extending the cone over an analytic subset of projective space across the origin.\nBishop's theorem replaces the dimension condition on $E$ by a volume condition on $A$, and Shiffman's theorem by a Hausdorff measure condition on $E$.\n",
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