# Boundary correspondence, principle of

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 [1]). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see [2]).
 [1] M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian) [2] L.D. Kudryavtsev, "On differentiable mappings" Dokl. Akad. Nauk SSSR , 95 : 5 (1954) pp. 921–923 (In Russian) [3] S.I. Pinchuk, "Holomorphic equivalence of certain classes of domains in $\CC^n$" Math. USSR-Sb. , 111 (153) : 1 (180) pp. 67–94; 159 Mat. Sb. , 111 (153) : 1 (1980) pp. 67–94; 159