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  "body": "\n## Statement\n\nLet $E \\subseteq \\C$ be a [[closed set|0G5AV9Q4RR]], and let $\\hat\\C = \\C \\cup \\set{\\infty}$ be the [[Riemann sphere|0CDMXGPSES]].\nThe following are equivalent.\n\n1. Let $f \\colon E \\to \\C$ be any function that is [[continuous|0P27W9RW0G]] on $E$ and [[holomorphic|0DG0Z72CQD]] on the [[interior|0C38S0KMJK]] of $E$, and let $\\varepsilon \u003e 0$ be arbitrary.\nThen there is an [[entire function|0419794R0T]] $g$ with $\\abs{f(z) - g(z)} \u003c \\varepsilon$ for all $z \\in E$.\n2. The set $\\hat\\C \\setminus E$ is [[connected|0D939CMME7]] and locally connected.\n\n## Remarks\n\nFor compact $E$, the set $\\hat\\C \\setminus E$ is an open subset of the sphere, so it is automatically locally connected, and connectedness of $\\hat\\C \\setminus E$ is the same as connectedness of $\\C \\setminus E$.\nSo for compact $E$ the theorem reduces to [[Mergelyan's theorem|5KKCSZJC29]], since entire functions can be approximated by polynomials on compact sets.\nThe new condition for unbounded $E$ is local connectedness of $\\hat\\C \\setminus E$ at the point $\\infty$.\n\nExample: let $E = \\R$.\nThen $\\hat\\C \\setminus \\R$ is the union of the two open half-planes and the point $\\infty$, which is connected and locally connected.\nSo every continuous function on $\\R$ is a uniform limit on all of $\\R$ of entire functions, a result first proved by Carleman.\n\nArakelyan also proved a version on an arbitrary open set $\\Omega$ in place of $\\C$, with $E$ closed in $\\Omega$.\nThere the approximants are holomorphic on $\\Omega$, and the one-point compactification of $\\Omega$ replaces $\\hat\\C$.\n\n## Explanations\n\n### L1 human\n\nMergelyan's theorem tells when functions on a compact set can be approximated by polynomials.\nArakelyan's theorem does the same for closed sets that may go off to infinity, with entire functions instead of polynomials.\nThe set must have no holes, and in addition it must not wind around near infinity in a way that traps parts of its complement.\nFor the real line, the conclusion is that any continuous function can be uniformly approximated by entire functions on the whole line.\n\n### L2 human\n\nThe theorem is the final answer to uniform holomorphic approximation on closed subsets of the plane.\nIt is used to construct entire functions with prescribed behaviour along unbounded sets, for example functions that tend to $0$ along every ray but are not identically $0$.\nIts proof combines Mergelyan-type estimates with a careful exhaustion of $E$ by compact pieces, where local connectedness at $\\infty$ controls how the pieces fit together.\n",
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