The confusion usage of atlas and maximal atlas I'm studying Loring Tu's _An Introduction to Manifolds_. In p.60 he said $, 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?

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.