# Jordan curve theorem

###### Theorem 1.

If $\Gamma$ is a simple closed curve in $\mathbb{R}^{2}$, then $\mathbb{R}^{2}\setminus\Gamma$ has precisely two connected components    (http://planetmath.org/ConnectedSpace).

###### Theorem 2.

If $\Gamma$ is a simple closed curve in the sphere $S^{2}$, then $S^{2}\setminus\Gamma$ consists of precisely two connected components.

###### Theorem 3.

Let $h:\mathbb{R}\to\mathbb{R}^{2}$ be a one-to-one continuous map such that $|h(t)|\to\infty$ as $|t|\to\infty$. Then $\mathbb{R}^{2}\setminus h(\mathbb{R})$ consists of precisely two connected components.

The two connected components mentioned in each formulation are, of course, the inside and the outside the Jordan curve, although only in the first formulation is there a clear way to say what is out and what is in. There we can define “inside” to be the bounded    connected component, as any picture can easily convey.

Title Jordan curve theorem JordanCurveTheorem 2013-03-22 13:08:53 2013-03-22 13:08:53 rmilson (146) rmilson (146) 9 rmilson (146) Theorem msc 54D05 msc 54A05