difference between maximal atlas and atlas in differential manifold i am reading the definition of atlas in differential manifold from U.C De book.. we know differential structure on a manifold is maximal atlas. but i want to know a counter example to distinguish between maximal atlas and just atlas.

The identical map $I$ from $\mathbb{R}$ to $\mathbb{R}$ forms an atlas by itself. However, $\\{I, 2 \cdot I \\}$ is also an atlas on that manifold. The maximal atlas based on this atlas would be the set of all diffeomorphisms from open subsets to open subsets of $\mathbb{R}$.