fundamental group of manifold, Lee's text topological manifold I am reading Lee' text "Introduction to Topological Manifold", I have a question about his proof of theorem 7.21.  I include his proof below for reference. 

My question is about the statement underlined in red.  I know that $U$ and $U'$ are connected since they are coordinate balls but how do we know that their intersection cannot be uncountable?  I couldn't think of a proof to show that they are countable.  Any help would be great.  Thank you. 

An open set of $R^n$ does not have an uncountable family of disjoint open subsets, because it is separable.
