Jordan Curve Image Equals Set Homeomorphic to Circle

Theorem

Let $C \subseteq \R^2$ be a subset of the Euclidean space $\R^2$.

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

Let $\tau_C$ and $\tau_{\mathbb S^1}$ be the subspace topologies on $C$ respectively $\mathbb S^1$ induced by the Euclidean topology on $\R^2$.

Then $\struct{ C, \tau_C }$ is homeomorphic to $\struct { \mathbb S^1 , \tau_{\mathbb S^1} }$, if and only if there exists a Jordan curve $\gamma : \closedint 0 1 \to \R^2$ with image equal to $C$.

Proof

By definition of simple loop, a Jordan curve is a simple loop in $\R^2$.

Euclidean Space is Complete Metric Space and Metric Space is Hausdorff shows that $\R^2$ is a Hausdorff space.

The result now follows from Simple Loop Image Equals Set Homeomorphic to Circle.

$\blacksquare$