Attainable boundary arc

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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 $|z| = 1$.


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


[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)
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Attainable boundary arc. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article