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Suppose $G$ is an open set in the complex plane and $f$ is a (complex valued) meromorphic function on $G$. IS $f$ necessarily an open map?

What if $f$ is an extended complex valued function? Is it an open map?

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Of course it suffices to consider the case where $f$ is non-constant. Then away from the poles, $f$ is holomorphic and hence an open map. And if we consider a one-point compactification $\hat{\Bbb{C}} = \Bbb{C}\cup\{\infty\}$, then again $f$ is an open map since it defines a holomorphic function between two Riemann surfaces $G$ and $\hat{\Bbb{C}}$. – Sangchul Lee Nov 20 '12 at 19:19

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