# simple boundary point

###### Definition.

Let $G\subset \u2102$ 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 $\gamma :[0,1]\to \u2102$ 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 such
that every point in the boundary of $G$ is a simple boundary point, then $\mathrm{\partial}\mathit{}G$ is a Jordan curve.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.

Title | simple boundary point |
---|---|

Canonical name | SimpleBoundaryPoint |

Date of creation | 2013-03-22 14:23:23 |

Last modified on | 2013-03-22 14:23:23 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 30-00 |

Classification | msc 54-00 |