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.)