Jordan Curve Theorem

Theorem
Let $\gamma: \left[{0 \,.\,.\, 1}\right] \to \R^2$ be a Jordan curve.

Let $\operatorname{Im} \left({\gamma}\right)$ denote the image of $\gamma$.

Then $\R^2 \setminus \operatorname{Im} \left({\gamma}\right)$ is a union of two disjoint connected components.

Both components are open in $\R^2$, and both components have $\operatorname{Im} \left({\gamma}\right)$ as their boundary.

One component is bounded, and is called the interior of $\gamma$.

The other component is unbounded, and is called the exterior of $\gamma$.