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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. enter image description here

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.

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

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    Simpler : it contains a point with coordinates in $\bf Q^n$, so you have a injective map from the set of components to $\bf Q^n$ – 2017-01-22
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    @Thomas, yes that's the proof that it is separable. It is reasonable to assume, though, that someone that's gotten to the point of knowing what the fundamental group of a general manifold is knows what *separable* means and how to prove that open sets hve that property... – 2017-01-22