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.
Solution 1:
An open set of $R^n$ does not have an uncountable family of disjoint open subsets, because it is separable.