User:Anghel/Sandbox

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 in $\R^2$.

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

Let $\phi { \restriction_X }$ denote the restriction of a function $\phi$ to a set $X$.

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


 * $ \phi {\restriction_{ \Img \gamma } } : \Img \gamma \to \mathbb S^1$ is a homeomorphism.


 * $ \phi {\restriction_{ \Int \gamma } } : \Int \gamma \to \map {B_1} { \mathbf 0 }$ is a homeomorphism.


 * $ \phi {\restriction_{ \Ext \gamma } } : \Ext \gamma \to \R^2 \setminus \map {B_1^-} { \mathbf 0 }$ is a homeomorphism.