Jordan-Schönflies Theorem
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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 $\R^2 \setminus \map {B_1^-} { \mathbf 0 }$.
Proof
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Source of Name
This entry was named for Marie Ennemond Camille Jordan and Arthur Moritz Schönflies.
Sources
- 1992: Carsten Thomassen: The Jordan-Schönflies Theorem and the Classification of Surfaces (Amer. Math. Monthly Vol. 99, no. 2: pp. 116 – 131) www.jstor.org/stable/2324180