# simple boundary point

###### Definition.

Let $G\subset{\mathbb{C}}$ be a region and $\omega\in\partial G$ (the boundary of $G$). Then we call $\omega$ a simple boundary point if whenever $\{\omega_{n}\}\subset G$ is a sequence converging to $\omega$ there is a path $\gamma\colon[0,1]\to{\mathbb{C}}$ such that $\gamma(t)\in G$ for $0\leq t<1$, $\gamma(1)=\omega$ 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 $G\subset{\mathbb{C}}$ is a bounded simply connected region such that every point in the boundary of $G$ is a simple boundary point, then $\partial G$ is a Jordan curve.

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title simple boundary point SimpleBoundaryPoint 2013-03-22 14:23:23 2013-03-22 14:23:23 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 30-00 msc 54-00