09 - Jordan Curve Theorem

Jordan Curve Theorem

Let $\gamma$ be a simple closed curve then

$\mathbb{C}\setminus {\gamma }^{\ast }$ has precisely one bounded and one unbounded component

1. Interior of $\gamma$: the bounded component (inside region of $\gamma$)
2. Exterior of $\gamma$: the unbounded component (outside region of $\gamma$)

Note that the components refer to the regions where $\gamma$ cuts $\mathbb{C}$