Open mapping theorem (function theory)

This article deals with the theorem from the mathematical branch of function theory. For more sentences of this name see the sentence of the open figure .

The theorem of the open mapping , sometimes also the openness theorem, is a theorem of function theory and states that images of open sets under holomorphic maps that are not constant on any connected component of the open set are open again. A consequence of this theorem is the maximum principle for holomorphic functions. Higher-dimensional statements of this kind do not apply.

Open map theorem for holomorphic functions

Be open and a holomorphic function that is not constant on any connected component of. Then there is an open crowd. ${\displaystyle U\subseteq \mathbb {C} }$ ${\displaystyle f\colon U\to \mathbb {C} }$ ${\displaystyle U}$ ${\displaystyle f(U)}$

A direct corollary is the domain loyalty of holomorphic functions.

Let be a non-constant, holomorphic function on a domain , then is a domain too. ${\displaystyle f\colon G\to \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle f(G)}$

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• George Marinescu: Function Theory . Lecture notes 2009, University of Cologne, pp. 41–42
• Eberhard Freitag, Rolf Busam: Function Theory 1 . Springer 2006, ISBN 9783540317647 , p. 123 ( limited online version in the Google book search)