# Critical values of a map to a Riemann surface

Let $f:X\to S$ be open proper holomorphic map, $X$ a complex manifold and $S$ a Riemann surface. Is it then true that the critical values $C\subset S$ of $f$ are a discrete supset?

So far I only noted that this would be true, if the set of critical points $K\subset X$ was discrete. But I don't know how to prove it. If $X$ was also a Riemann surface, I could apply the identity principle to $f'$, but in the general case I don't know how to proceed.

Edit: I know about Sard's theorem. I don't see why it is strong enough though.

• en.wikipedia.org/wiki/Sard%27s_theorem – Alan Muniz Jun 24 '18 at 16:56
• @AlanMuniz: In the smooth category, the set of critical values can be dense, in fact (still of measure 0). – Ted Shifrin Jun 24 '18 at 16:58
• Sure... The fact that the map is proper and holomorphic is crucial. – Alan Muniz Jun 24 '18 at 17:08
• In this case, if $A\subset X$ is an analytic subvariety, then $f(A) \subset S$ will be an analytic subvariety. See page 162 of Gunning & Rossi book Analytic Functions of Several Complex Variables. – Alan Muniz Jun 25 '18 at 11:28
• I don't have access to the book mentionned by Alan Muniz, but I think he is refering to Remmert's theorem (if $f: X \to Y$ is proper holomorphic, then the image of an analytic set is an analytic subset of $Y$). – Glougloubarbaki Jun 25 '18 at 11:52