Definition:Interior of Simple Closed Contour

Definition

Let $C$ be a simple closed contour in the complex plane.

Let $f : \closedint 0 1 \to \R^2$ be a Jordan curve.

Let $\phi : \R^2 \to \C$ be defined by:

$\map \phi {x, y} = x + i y$

Let $\Img C = \map \phi {\Img f}$, where $\Img C$ denotes the image of $C$, and $\Img f$ denotes the image of $f$.

Then the interior of $C$ is denoted $\Int C$ and defined as:

$\Int C = \map \phi {\Int f}$

where $\Int f$ denotes the interior of $f$.

Also see

• Interior of Simple Closed Contour is Well-Defined
• Complex Plane is Homeomorphic to Real Plane, which shows that $\phi$ is a homeomorphism between $\R^2$ and $\C$.

Sources

• 2001: Christian Berg: Kompleks funktionsteori: $\S 2.2$