Standard Smooth Strucutre On Euclidean Spaces Confusion

Question

I am confused as to how $\mathscr A=\{(\mathbb R^n,id)\}$ forms a smooth structure on $\mathbb R^n$.

Here's why.

Let $U$ be the open ball of unit radius centered at origin of $\mathbb R^n$ and define $\varphi:U\to U$ as $\varphi(x)=x$ for all $x\in U$.

Then $(U,\varphi)$ is a chart on $\mathbb R^n$.

Further, $(U,\varphi)$ is smoothly compatible with $(\mathbb R^n,id)$.

But then $(U,\varphi)$ should lie in $\mathscr A$, which is doesn't.

What am I doing wrong?

Answer 1

$A$ is an atlas of one chart. The smooth structure induced by $A$ is $A$'s equivalence class. We call two atlaces equvalent iff their union is an atlas.

Edit: In terms of your texbook, they mean the atlas you get when you include all compatible charts.

Answer 2

$\mathscr A$ is an atlas but no a maximal atlas.