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  "body": "\n## Statement\n\nLet $C \\subset \\R^2$ be a Jordan curve, that is, the image of a [[topological embedding|00ZG9Y9TR4]] $\\gamma \\colon S^1 \\to \\R^2$ of the [[circle|060W5CBZP6]] into the [[plane|0XPR1E85W0]].\nThen $\\R^2 \\setminus C$ has exactly two [[connected components|0V032VWTHD]].\nOne component is bounded and the other is unbounded.\nThe curve $C$ is the boundary of each component.\n\n## Remarks\n\nFor a polygon the theorem is elementary: the parity of the number of crossings of a ray with the polygon tells the two sides apart.\nFor a general curve, one proof approximates by polygons; another computes with homology, where Alexander duality gives $\\tilde H_0(\\R^2 \\setminus C) \\cong \\Z$, so there are two components.\nThe theorem is hard because a Jordan curve can be very irregular, for example nowhere differentiable, or of positive area as an Osgood curve.\nThe bounded component is called the interior of $C$ and the unbounded one the exterior.\nThe same holds on the sphere $S^2$, without the words bounded and unbounded.\n\n## Explanations\n\n### L1 human\n\nA closed loop drawn in the plane without crossing itself cuts the plane into an inside and an outside.\nTo go from a point inside to a point outside, a path must cross the loop.\nThis seems obvious, but the loop can be extremely wiggly, so the proof is subtle.\n\n### L2 human\n\nThe theorem is the first case of the Jordan–Brouwer separation theorem: an embedded $(n-1)$-sphere in $\\R^n$ separates it into two components.\nIts strengthening in the plane, the Schoenflies theorem, says that the inside is a topological disk.\nIn higher dimensions this strengthening fails without extra hypotheses, as the Alexander horned sphere shows.\nIn complex analysis, the theorem gives meaning to \"the inside of a contour\" in Cauchy's integral theorem.\n",
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