Ramification points of holomorphic map is a closed subset

Question

I am referring to this answer : https://math.stackexchange.com/a/273269/1162293

I do not see why the argument let us tell that ramification points are a closed subset of $X$. Has someone some more details to give ?

Answer 1

If $f: X \to Y$ is not ramified at $x \in X$, then it is a local homeomorphism around $x$. I.e., there is an open neighborhood $x \in U \subseteq X$ s.t. $f|_U$ is a local homeomorphism. Hence $f$ is unramified at all points of $U$.

That shows that the complement of the ramified points is open.