Open Mapping Theorem (complex Analysis)

Open Mapping Theorem (complex Analysis)

In complex analysis, the open mapping theorem states that if U is a domain of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. it sends open subsets of U to open subsets of C).

The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the real line, for example, the differentiable function f(x) = x2 is not an open map, as the image of the open interval (−1, 1) is the half-open interval [0, 1).

The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of any real line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.