| context | $\langle M,T\rangle$ … second-countable Hausdorff space |
| context | $n\in \mathbb N$ |
| definiendum | $\langle M,A\rangle\in$ it |
| postulate | $A$ maximal in smooth atlas($\langle M,T\rangle,n$) |
Effectively, a smooth manifold would be given by providing any atlas. But then, due to the redundancy of some charts on small open sets, different atlases give rise to equivalent mathematical objects and so a smooth manifold is defined as the biggest and hence unique one amongst those objects.
Wikipedia: Differentiable manifold