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# On zeroes and poles of a meromorphic function

Let be a meromorphic function on a (connected) Riemann Surface . Show that the zeros and the poles of are isolated points.

Solution

Let be a Riemann surface and let be a meromorphic function on . Then an be considered as a holomorphic map which is not identically equal to . But for holomorphic maps between Riemann Surfaces the Identity Theorem holds.
• If the set of zeros of contained a limit point, then, by the Identity Theorem, should be equal to but we have assumed that is not constant.
• If the set of poles of contained a limit point, then, by the Identity Theorem, should be equal to . But we have excluded that case by definition of a meromorphic function.

Hence, the zeros and the poles of are isolated points.

Remark: Using the Identity Theorem we can also prove that the set of ramification points of a proper, non-constant, holomorphic map between Riemann Surfaces consists only of isolated points.

The exercise can also be found at mathimatikoi.org