# Is the maximal atlas for a topological manifold unique?

I'm reading "An Introduction to Manifolds" by Tu. In this book, the definition of a topological manifold is a Hausdorff, second countable locally Euclidean space and the definition of a smooth manifold is a topological manifold with a maximal (pairwise $C^\infty$-compatible) atlas.

I know that given a topological manifold $M$ and an atlas $\mathfrak{A}$ on $M$, there is a unique maximal atlas on $M$ that contains $\mathfrak{A}$. My question is that is it possible to have two distinct maximal atlases on the topological manifold $M$ (that is,they are not required to contain a given atlas on $M$)?

• If I understand correctly, then Proposition 5.10 of Tu may be understood in the following way as illustrated in the answer by Alex Ortiz. If $A_1$ and $A_2$ are two atlases which are not compatible with each other, then they are contained in two different maximal atlases $\mathcal{M}_1$ and $\mathcal{M}_2$, where $\mathcal{M}_1 \neq \mathcal{M}_2$. Commented Jun 10, 2020 at 16:43

In general there is not a unique maximal atlas on the manifold $$M$$. Consider $$\Bbb R$$ and the charts $$(\Bbb R,\operatorname{Id})$$ and $$(\Bbb R,f)$$, where $$f(x) = x^3$$. Each of these charts cover $$\Bbb R$$, but it's easy to check they are not smoothly compatible, so they generate distinct maximal atlases. However, the smooth manifolds we get by equipping $$\mathbb R$$ with each of these maximal atlases are diffeomorphic.