Jordan Curve Characterization of Simply Connected Set

Theorem
Let $D \subseteq \R^2$ be a path-connected subset of the Euclidean plane.

Then $D$ is simply connected, the following condition holds:


 * For all Jordan curves $f : \closedint 0 1 \to \R^2$ with $\Img f \subseteq D$, we have $\Int f \subseteq D$.

Here $\Img f$ denotes the image of $f$, and $\Int f$ denotes the interior of $f$.