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  "body": "\n## Statement\n\n1. Let $C \\subset \\R^2$ be a [[Jordan curve|687RGCHBTP]]. Then there is a [[homeomorphism|0QT00PB5G2]] $h \\colon \\R^2 \\to \\R^2$ with $h(C) = S^1$, the unit circle. In particular, the closure of the bounded component of $\\R^2 \\setminus C$ is homeomorphic to the [[closed disk|0Q0H13G8ZB]] $D^2$.\n2. (Generalized Schoenflies theorem.) Let $n \\ge 2$ and let $e \\colon S^{n-1} \\times [-1, 1] \\to S^n$ be a [[topological embedding|00ZG9Y9TR4]], where $S^n$ is the [[$n$-sphere|T62369RFH6]]. Let $\\Sigma = e(S^{n-1} \\times \\set{0})$. Then $S^n \\setminus \\Sigma$ has two components, and the closure of each is homeomorphic to the closed ball $D^n$.\n\n## Remarks\n\nThe hypothesis in part 2 says that the sphere $\\Sigma$ is bicollared.\nFor a codimension-one sphere this is equivalent to being locally flat.\nIt cannot be dropped for $n = 3$: the Alexander horned sphere is an embedded $2$-sphere one of whose complementary regions is not simply connected, so its closure is not a ball.\nIn the plane no such hypothesis is needed, because every Jordan curve is tame.\nOne proof of part 1 uses the Riemann mapping theorem and Carathéodory's extension theorem for the interior; part 2 was proved by Morton Brown in 1960.\nFor smooth embedded spheres in $\\R^4$, whether the closed inside is always a smooth ball is open.\n\n## Explanations\n\n### L1 human\n\nThe Jordan curve theorem says a loop in the plane has an inside and an outside.\nThe Schoenflies theorem says more: the inside, with the loop added, is just a deformed disk.\nOne can straighten the whole plane so that the loop becomes a round circle.\n\n### L2 human\n\nIn higher dimensions, the naive analogue fails: there are wild embeddings of $S^2$ in $\\R^3$ whose outside is not the outside of a ball.\nThe generalized theorem shows that all such wildness lives at points where the sphere is not locally flat.\nIt is a basic tool in topological manifold theory, for example for the annulus theorem and for showing that the connected sum is well defined.\n",
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