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.
On
This is because $\mathbb{R}^{n}$ is second countable (and thus is any its open subset $S$). In particular, every open cover of $S$ has a countable subcover (this is a basic property of second countable spaces).
Connected components of $S$ form its open cover. Would there be uncountably many of them, there would be no its countable subcover. Hence any open subset $S \subseteq \mathbb{R}^{n}$ must have at most countably many connected components. Apply this to $S = U \cap U'$.
An open set of $R^n$ does not have an uncountable family of disjoint open subsets, because it is separable.