The Open Mapping Theorem of complex analysis states that a nonconstant holomorphic function on a connected open subset of the complex plane maps every open set to an open set. A direct consequence of complex differentiability rather than a separately assumed hypothesis, it explains, among other things, why the maximum modulus of a nonconstant holomorphic function cannot occur at an interior point, since an interior local maximum of the modulus would force nearby image points outside an open neighborhood.
Facts
StatementIf U is a domain of the complex plane and f is a non-constant holomorphic function on U, then f is an open map, sending open subsets of U to open subsets of the complex plane. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Sources
1. Open mapping theorem (complex analysis) (Wikipedia)
Intro, sentence 1Quote, Intro, sentence 1
the open mapping theorem states that if
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