the open mapping theorem

Take any little patch of the plane — an open disk, say — and push it through a holomorphic function. The open mapping theorem promises the result is again an open set: a genuinely two-dimensional blob with no sharp edge or pinched boundary that the patch suddenly collapses onto. A non-constant holomorphic function can stretch, twist, and bend regions, but it can never crush an open set down to a curve or a single point. Openness in, openness out.

The statement: if f is holomorphic and non-constant on a region (a connected open set), then for every open subset U the image f(U) is open. Why this happens: near a point z_0 where f(z_0) = w_0, the function looks locally like w_0 plus a power (z - z_0)^m times a nonvanishing factor (m being the order of the zero of f - w_0 at z_0). Taking an m-th root locally, f behaves near z_0 like the map z mapsto z^m composed with a conformal change of coordinates — and z mapsto z^m is an open map (it sends small disks onto small disks, wrapping m times). So small disks around z_0 map onto sets containing small disks around w_0, which is exactly openness. The argument principle gives the quantitative version: for w near w_0, the equation f(z) = w has exactly m solutions near z_0, so values near w_0 are genuinely attained.

This single structural fact forces the maximum-modulus principle almost for free. If |f| had an interior maximum at z_0, then f(z_0) would be a boundary point of the image f(U) — the value of largest modulus, sitting on the outer edge of the image blob. But the open mapping theorem says f(U) is open, so it has no boundary points inside it; hence f(z_0) cannot be an interior point of the image, a contradiction unless f is constant. The theorem is also why holomorphic functions are so geometrically rigid. The essential caveat lives in the words: f must be non-constant — a constant function maps everything to a single point, which is not open.