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On zeroes and poles of a meromorphic function

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

Solution

Let X be a Riemann surface and let f be a meromorphic function on X. Then f an be considered as a holomorphic map f:X \rightarrow \hat{\mathbb{C}} which is not identically equal to \infty. But for holomorphic maps between Riemann Surfaces the Identity Theorem holds.
  • If the set of zeros of f contained a limit point, then, by the Identity Theorem, f should be equal to 0 but we have assumed that f is not constant.
  • If the set of poles of f contained a limit point, then, by the Identity Theorem, f should be equal to \infty. But we have excluded that case by definition of a meromorphic function.

Hence, the zeros and the poles of f 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

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