# Boundary correspondence, principle of

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A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping $f$ if the facts that $f$ is a continuous mapping of the closure $\overline{G}\;$ of a domain $G$ onto the closure $\overline{D}\;$ of a domain $D$ and $f$ is a homeomorphism of $\overline{G}\; \setminus G$ onto $\overline{D}\; \setminus D$ imply that $f$ is a topological mapping of $\overline{G}\;$ onto $\overline{D}\;$. Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. Boundary correspondence (under conformal mapping)).

If $G$ and $D$ are plane domains with Euclidean boundaries homeomorphic to a circle and $D$ is bounded, then the principle of boundary correspondence holds for analytic functions $f$ on $G$, i.e. $f$ is a conformal mapping of $G$ onto $D$. In addition to that given above, various other forms of the principle of boundary correspondence are commonly used for conformal mappings (see ). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see ).

How to Cite This Entry:
Boundary correspondence, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_correspondence,_principle_of&oldid=46128
This article was adapted from an original article by B.P. Kufarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article