An Uncountable Subset of a Topological Space with a Countable Base $\Rightarrow$ M contains a limit point Please provide hints and not a direct answer as I would like to figure this out myself.
Prove that if $M$ is an uncountable subset of a topological space with a countable base, then some point of $M$ is a limit point of $M$
So far, If M has a countable base then there exits a countable everywhere dense subset $N \subset M$ such that $[N]=M$ $\Rightarrow$ $M$ contains all the limit points of $N$. The only way for our claim to be false is if N is made entirely of interior points $\Rightarrow$ N is open. Then $M$ is open is this a contradiction? Any hints on how to proceed would be greatly welcomed. 

Hint: if $x \in M$ is not a limit point of $M$, then there is an open set $A_x$ in the given base such that $A_x \cap M = \{x\}$. If no point of $M$ is a limit point of $M$, what can you say about the sets $A_x$?
