# Attainable boundary arc

*of a domain $G$ in the $z$-plane*

A Jordan arc forming part of the boundary of $G$ and at the same time forming part of the boundary of some Jordan domain $g \subset G$. Each point on an attainable boundary arc is an attainable (from the inside of $g$) boundary point of $G$ (cf. Attainable boundary point). A conformal mapping of a simply-connected domain $G$ onto the unit disc $D = \{ z : |z| < 1 \}$ can be continuously extended to the non-terminal points of an attainable boundary arc to a homeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$.

#### Comments

The standard Western terminology is accessible boundary arc and accessible boundary point, see e.g. [a1].

#### References

[a1] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9 |

[a2] | A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 2 (Translated from Russian) |

**How to Cite This Entry:**

Attainable boundary arc.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Attainable_boundary_arc&oldid=36924