Jordan curve theorem

Remarks

For a polygon the theorem is elementary: the parity of the number of crossings of a ray with the polygon tells the two sides apart. For 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. The theorem is hard because a Jordan curve can be very irregular, for example nowhere differentiable, or of positive area as an Osgood curve. The bounded component is called the interior of $C$ and the unbounded one the exterior. The same holds on the sphere $S^2$, without the words bounded and unbounded.

Going deeper

For a student

The theorem is the first case of the Jordan–Brouwer separation theorem: an embedded $(n-1)$-sphere in $\R^n$ separates it into two components. Its strengthening in the plane, the Schoenflies theorem, says that the inside is a topological disk. In higher dimensions this strengthening fails without extra hypotheses, as the Alexander horned sphere shows. In complex analysis, the theorem gives meaning to "the inside of a contour" in Cauchy's integral theorem.