Open mapping theorem (function theory)
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.
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.
swell
- 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)