Consider the definition of differentiable structure (Lectures on Differential Geometry, S.S. Chern):
Suppose $M$ is an m-dimensional manifold. If a given set of coordinate charts ${\mathcal A} = \{(U,\phi_U),(V,\phi_V),(W,\phi_W),\cdots\}$ on $M$ satisfies the following conditions, then we call ${\mathcal A}$ a $C^r$-differentiable structure on $M$:
- $\{U,V,W,\cdots\}$ is an open covering of $M$;
- any two coordinate charts in ${\mathcal A}$ are $C^r$-compatible;
- ${\mathcal A}$ is maximal, i.e., if a coordinate chart $(\tilde{U},\phi_{\tilde{U}})$ is $C^r$-compatible with all coordinate charts in ${\mathcal A}$, then $(\tilde{U},\phi_{\tilde{U}})\in{\mathcal A}$.
An example for this definition in that book, is as following: For $M={\mathbb R}$, let $U=M$, and $\phi_U$ be the identity map. then $\{(U,\phi_U)\}$ is a coordinate covering of ${\mathbb R}$. This provides a smooth differentiable structure on ${\mathbb R}$, called the standard differentiable structure of ${\mathbb R}$.
I don't understand how $\{(U,\phi_U)\}$ provides a smooth differentiable structure. Is it maximal? Consider for example $\{(U,\phi_U),(V,\phi_V)\}$ where $V=(0,1)$ and $\phi_V$ the identity map. This puzzles me.