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I'm studying Loring Tu's An Introduction to Manifolds. In p.60 he said enter image description here

Given a smooth manifold $M=(\underline{M},\Phi_{\text{maxi}})$, it is understood by people that there exist a maximal atlas $\Phi_{\text{maxi}})$ of that underlying set $\underline{M}$. However, what does the "atlas" refer to in (ii)? Is it belongs to the origin maximal atlas (given immediately when he said "Let $M$ be a manifold ..."), or could be any atlas even outside the original maximal atlas? How to tell from the context?

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  • $\begingroup$ Another viewpoint: is you can cover $M$ by a subset of charts of the maximal atlas $(U,\phi)$ s.t. the compositions $f\circ\phi^{-1}$ are smooth, then the same is true for all the charts of the maximal atlas. $\endgroup$ – Martín-Blas Pérez Pinilla Aug 1 '19 at 10:05
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Isn't required that the atlas in (ii) be maximal, but a such atlas can be extended to a maximal atlas preserving the property, namely the maximal atlas defining the smooth structure.

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  • $\begingroup$ Is the atlas in (ii) require to be $C^{\infty}$ equivalent with the original maximal atlas? (To be concrete, it is possible for a topological space $X$ to have several different maximal atlases. When giving a smooth manifold, it is known that a maximal atlas is specified. Then is the atlas mentioned in (ii) belongs to it?) $\endgroup$ – Eric Aug 1 '19 at 10:06
  • $\begingroup$ @Eric, yes. See my other comment. $\endgroup$ – Martín-Blas Pérez Pinilla Aug 1 '19 at 10:43

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