# Open mapping theorem for Riemann surfaces

What restriction must one impose on a Riemann surface M in order for all biholomorphic $f:M\to\mathbb{C}$ to be open mappings, aka mappings of $M$ onto open subsets $f(M)\subset\mathbb{C}$?

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The open mapping theorem from complex analysis carries over to Riemann surfaces basically immediately. – Jack Huizenga Jul 16 '11 at 0:03

Clarification: If you don't require Riemann surfaces to be connected then the correct statement would be: A holomorphic map $f : X \rightarrow Y$ between Riemann surfaces is an open map provided that it is not constant on any connected component of $X$.