Can it be proved that a Meromorphic function only has a countable number of poles? One definition I have seen is that a Meromorphic function has at most a countable number of poles. Another says that a function f is Meromorphic if every point is either a pole or the function is analytic there. 
Now first an easy question, but I am unsure of it:
1. That poles are isolated, that is usually taken as a definition of the pole?
Now come the hard question:
2. Can it be proved that if we use definition 2 of a Meromorphic function it can at most have a coutnable number of poles? If the answer to question 1 is yes, I guess we can assume for contradiction that it has more than a countable number of poles, and then show that it must have a limit point, and hence not be isolated? Is this hard to prove?
PS: I think an equivalent question for 2 in terms of $\mathbb{R}^2$ is that lets say that the set A is bigger than just beeing countable(I don't know if this implies if it is uncountable?). Then the set A must have a sequence of distinct points, with a limit point in $\mathbb{R}^2$. But I don't think the limit point must be in A? Is this something we can prove?
 A: Poles are by definition isolated singularities, so there is an open disk of positive radius around each of them in which no other singularity occurs.  This is sufficient to prove there are at most countably many poles in the complex plane (or in some open domain in the complex plane).  This can be seen as a corollary to no uncountable sum of positive real numbers (the areas of these disks) can be finite.
A: Ad 1. Yes, it is part of the definition or a consequence of the definition that a pole is an isolated singularity of the function.
Ad 2. Yes, since poles are isolated singularities, a meromorphic function can have at most countably many poles. That is not hard to prove: Every compact subset $K$ of the domain of the function can only contain finitely many poles [otherwise the set of poles would have a limit point in $K$], and every open subset of $\mathbb{C}$ (or $\widehat{\mathbb{C}}$) is the union of countably many compact subsets (since it is locally compact and second countable), so the set of poles is a countable union of finite sets.
