TITLE: Proofs given in undergrad degree that need Continuum hypothesis?
QUESTION [20 upvotes]: Or alternative you need to assume CH is false.
I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non commutative algebra course I'm studying. 
However, I have yet to see CH be used to prove stuff. 
Anyone give a theorem that comes up in undergrad degree like last year of it that uses CH or not CH is used? 
As personally, I can't think where I've seen some sort of argument that assumes say CH or looks like you are assuming it. The same can't be said about the axiom of choice, which in about 50% of the proofs.  
CH scares me abit. As I can't see how it can be independent.

REPLY [3 votes]: As a simple implication of CH, I can say that in general topology, M. E. Rudin proves that if CH is true then $\mathbb{R}^\omega $ with box topology is a normal topological space. [General topology and its applications, vol.2, 1972]