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(1), In the proof of Lemma 1.10: Every topological manifold has a countable basis of precompact coordinate balls, what is the function of $B_{r^{'}}(x)$? Why do we need it? Proof of Lemma 1.10

(2), I don't understand the definition of 'regular coordinate ball'. I mean, why do we require the condition $\varphi(\overline{B})=\overline{B}_r(0)$ in addition to $\varphi({B})={B}_r(0)$ and $\varphi(B')=B_{r^{'}}(0)$?

Definition of regular coordinate ball

(3), In the example 1.26 (Open submanifolds), does the author mean $A_U$ is exactly the set of all $(V,\varphi|V)$? Example 1.26

(4), Some exercise ask me to prove something in the cases of smooth manifold without boundary or with boundary. But I usually get confused in the latter case because I think smooth manifold with boundary is much more complex than manifold without boundary. You need to consider much more things when doing the proof. Is there any trick or good way to think of these problems so that one doesn't need to consider case of manifold with nonempty boundary?

Thanks!

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