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Let $X$ and $Y$ be Riemann surfaces (not necessarily compact) and $f:X\to Y$ a holomorphic map. A point $x\in X$ is said to be a ramificiation point of $f$ if the multiplicity of $f$ at $x$ is $\geq 2$. The set of all ramification points of $f$ is a discrete subset of $X$. A point $y\in Y$ is said to be a branch point if it is the image of a ramification point. Why does the set of branch points form a discrete subset of $Y$?

(This is asserted in p.46 of Miranda's Algebraic Curves and Riemann Surfaces. If $X$ is compact then this is obvious because there are only finitely many ramification points. But I can't handle the general case.)

E. Kevin
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  • Use the theorem on the local form of a holomorphic map between Riemann surfaces. – Thorgott Jun 20 '21 at 13:54
  • For noncompact $X$ this is false. https://math.stackexchange.com/questions/3813927/set-of-branch-points-isnt-discrete-but-branch-points-are-isolated/3888954#3888954 – Moishe Kohan Jun 20 '21 at 20:28

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