Definition of Maximal Smooth Atlas: In the text GTM $218$ by John Lee, the author defines a smooth altas $\mathcal{A}$ to be a maximal smooth atlas if any chart that is smoothly compatible with every chart in $\mathcal{A}$ is already in $\mathcal{A}$.
The author makes a comment that, if a topological manifold $M$ can be covered by a single chart, the smooth compatibility condition is trivially satisfied, so any such chart automatically determines a smooth structure on $M$, which means a maximal smooth altas on $M$.
Now let's consider an example.
Take the manifold $M=\mathbb{R}^n.$ Let $\mathcal{A}_1=\{ (\mathbb{R}^n,Id_{\mathbb{R}^n})\}$ and $\mathcal{A}_2=\{ (B_1(x),Id_{B_1(x)}):x\in \mathbb{R}^n\}$.
By author's comment, $\mathcal{A}_1$ should be a maximal smooth atlas on $M$. But it is obvious that every chart in $\mathcal{A}_2$ is compatible with the only chart in $\mathcal{A}_1$. And by the definition of maximal smooth atlas, $\mathcal{A}_1$ is not maximal because at least it doesn't contain those charts in $\mathcal{A}_2$.
I'm quite confused now. Thanks for help.
Best Answer
No, I didn't say that $\mathscr A_1$ is a maximal smooth atlas, I said it determines a maximal smooth atlas. This is a reference to Proposition 1.17(a), which says "every smooth atlas $\mathscr A$ for $M$ is contained in a unique maximal smooth atlas, called the smooth structure determined by ${\mathscr A}$."