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I have a minor problem on understanding the definition. Say, in the book An Introduction of Manifolds by Loring Tu, he gave a definition as below. My question is: are the chart $(U,\phi)$ and $(V,\psi)$ mentioned in the definition belongs to the atlas of the given manifold $N$ and $M$? To be clear, we know that when we say "manifold $N$", we mean a underlying set together with a "maximal atlas $\Phi_N$," so is $\phi\in\Phi_N$ here? It's hard to guess from the context. enter image description here

Also, Prop 6.8(ii)(iii) is weird to me:

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Yes: if $M$ is a smooth manifold, then whenever you talk about a chart in $M$ (or on $M$, or of $M$, etc.) that always refers to a chart in the atlas of $M$ unless specified otherwise.

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  • $\begingroup$ but then the equivalence of ii and iii in the prop seems too trivial, and it is strange to state as two different sentences $\endgroup$
    – Eric
    Jul 13, 2019 at 17:45
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    $\begingroup$ The equivalence of (ii) and (iii) is not trivial. (iii) refers to every chart in the atlas of definition of $M$ (i.e., the maximal atlas) while (ii) refers to only charts in some atlas (which may be a proper subset of the maximal atlas). $\endgroup$ Jul 13, 2019 at 17:50

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