Example of two atlases on a manifold $M$ which give rise to different sets of smooth functions on $M$.
Lee states that there will be many possible atlases that give rise to the same smooth structures on a manifold $M$; that is they determine the same collection of smooth functions on $M$.
Unfortunately, I don't see how different atlases will give rise to different collections of smooth functions in the first place.
For example, I'm struggling to think of two atlases on $\mathbb{R}$ which would not give the same collection of smooth functions; namely the ones in the sense of ordinary calculus.
Could someone provide me with an example of a manifold and two atlases on this manifold such that the collection of smooth functions determined by the atlases are not equivalent?
Let $\{(U_\alpha,\phi_\alpha)\}_{\alpha\in\mathcal{A}}$ and $\{(U_\beta,\phi_\beta)\}_{\beta\in\mathcal{B}}$ be two smooth (that is $\mathcal{C}^{\infty}$) atlases on a topological manifold $M$. By definition the $\mathcal{C}^{\infty}$-structures defined by these two atlases are equivalent if their union $\{(U_\alpha,\phi_\alpha)\}_{\alpha\in\mathcal{A}\cup\mathcal{B}}$ is also a smooth atlas, that is, the transition maps between the charts of different atlases are smooth. It can be shown, that this definition, is equivalent to saying that two $\mathcal{C}^{\infty}$-structures are equivalent if they determine the same set of smooth functions $f:M\rightarrow\mathbb{R}$.
Here is an example of two non-equivalent $\mathcal{C}^{\infty}$-structures:
Consider the topological manifold $\mathcal{N} = \mathbb{R}$ equipped with the $\mathcal{C}^{\infty}$-structure determined by the $\mathcal{C}^{\infty}$-atlas consisting of the single chart $(\mathbb{R}, Id)$.
Next consider $\mathbb{R}$ equipped with the $\mathcal{C}^{\infty}$-structure determined by the $\mathcal{C}^{\infty}$-atlas consisting of the single chart $(\mathbb{R}, \phi)$ where $\phi(x) = x^{3}$.
We can now show, that each one of the above $\mathcal{C}^{\infty}$-atlases (and thus: their $\mathcal{C}^{\infty}$-structures) are non-equivalent (because, for example, the change of coordinates: $[Id\circ\phi^{-1}](x)=x^{1/3}$ is not smoothly differentiable at $0$), therefore, the $\mathcal{C}^{\infty}$-structures determined by them are different.