simple boundary point

Definition.

Let $G\subset ℂ$ be a region and $\omega \in \partial G$ (the boundary of $G$). Thenwe call $\omega$ a simple boundary point if whenever$\{{\omega }_n\}\subset G$ is a sequence converging to $\omega$ there is a path$\gamma :[0,1]\to ℂ$ such that$\gamma (t)\in G$ for , $\gamma (1)=\omega$ and there is a sequence$\{t_n\}\in [0,1)$ such that $t_n\to 1$ and $\gamma (t_n)={\omega }_n$ for all$n$.

For example if we let $G$ be the open unit disc, then every boundary point is a simple boundary point. This definition is useful for studying boundary behaviour of Riemann maps (maps arising from the Riemann mapping theorem), and one can prove for example the following theorem.

Theorem.

Suppose that $G\mathrm{\subset }\mathrm{C}$ is a bounded simply connected region suchthat every point in the boundary of $G$ is a simple boundary point, then $\mathrm{\partial }\mathit{}G$ is a Jordan curve.