Jordan-Schönflies Theorem

Theorem
Let $\gamma : \closedint 0 1 \to \R^2$ be a Jordan curve.

Let $\Img \gamma$ denote the image of $\gamma$, $\Int \gamma$ denote the interior of $\gamma$, and $\Ext \gamma$ denote the exterior of $\gamma$.

Let $\mathbb S^1$ denote the unit circle whose center is at the origin $\mathbf 0$ of the Euclidean space $\R^2$..

Let $\map {B_1} { \mathbf 0 }$ denote the open ball in $\R^2$ with radius $1$ and center $\mathbf 0$, and let $\map {B_1^-} { \mathbf 0 }$ denote the closed ball in $\R^2$ with radius $1$ and center $\mathbf 0$.

Then there exists a homeomorphism $\phi : \R^2 \to \R^2$ such that:


 * The restriction of $\phi$ to $\Img \gamma \times \mathbb S^1$ is a homeomorphism between $\Img \gamma$ and $\mathbb S^1$.


 * The restriction of $\phi$ to $\Int \gamma \times \map {B_1} { \mathbf 0 }$ is a homeomorphism between $\Int \gamma$ and $\map {B_1} { \mathbf 0 }$.


 * The restriction of $\phi$ to $\Ext \gamma \times \R^2 \setminus \map {B_1^-} { \mathbf 0 }$ is a homeomorphism between $\Ext \gamma$ and $\R^2 \setminus \map {B_1^-} { \mathbf 0 }$.