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

**Solution**

*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 *