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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $X \\subseteq \\R^{n+1}$ be a subset [[homeomorphic|0QT00PB5G2]] to the [[$n$-sphere|T62369RFH6]] $S^n$.\nThen the complement $\\R^{n+1} \\setminus X$ has exactly two [[connected components|0V032VWTHD]], one bounded and one unbounded, and $X$ is the [[boundary|05D5EGMFYV]] of each of them.\n\n## Remarks\n\nThe proof uses [[Alexander duality|KK05CSTM15]] in $S^{n+1} = \\R^{n+1} \\cup \\set{\\infty}$.\nIt gives $\\tilde{H}_0(S^{n+1} \\setminus X; \\Z) \\cong \\tilde{H}^n(X; \\Z) \\cong \\Z$, with Čech cohomology on the right.\nSo $S^{n+1} \\setminus X$ has two path components, which are open, hence also its connected components.\nRemoving the point $\\infty$ from the component that contains it does not disconnect it, since $n + 1 \\ge 2$.\nFor $n = 1$ the theorem is the Jordan curve theorem in the plane.\n\nExample: the unit sphere $S^n$ itself separates $\\R^{n+1}$ into the open unit ball and the set $\\set{x : \\norm{x} \u003e 1}$.\n\nFor $n \\ge 2$ the theorem says nothing about the shape of the components.\nFor the Alexander horned sphere in $\\R^3$, the unbounded component is not simply connected, unlike the outside of a round sphere.\nThe Schoenflies theorem, which says that the bounded component is an open disc, holds for $n = 1$ but needs extra hypotheses, such as local flatness, for $n \\ge 2$.\nThe case $n = 0$ is excluded: two points on the line $\\R$ leave three components.\n\n## Explanations\n\n### L1 human\n\nA closed surface in space without self-intersections, however crumpled, has an inside and an outside.\nTo get from the inside to the outside you must cross the surface, and every point of the surface touches both sides.\n\n### L2 human\n\nThe theorem is one of the first applications of homology to geometry.\nIt is used to prove [[invariance of domain|6X2D0JDFQW]], and through it the topological invariance of dimension.\nThe contrast with the Schoenflies problem shows that homology can detect the number of components but not their homeomorphism type.\n",
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