I want to explicitly specify a maximal smooth atlas on a $\mathbb{T}^2$. I am following John Lee - introduction to smooth manifolds. He writes that the product smooth manifold structure defines a smooth structure on $\mathbb{T}^n = \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$. To me it clear that the product smooth manifold structure it specifies a smooth atlas, but I cannot see that it is maximal. Can we see that this atlas is maximal, or specify another one explicitly which is maximal?

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    $\begingroup$ I don't think you need to show that it's maximal. There's a proposition saying that every smooth atlas is contained in a unique maximal atlas, unless I'm remembering incorrectly. $\endgroup$ – Joshua Ruiter Feb 12 '17 at 21:59
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    $\begingroup$ The product atlas you are considering is not necessarily maximal. If that were the case, all charts on $\mathbb{R}^2$ would have a coordinate domain of the form $U \times V$ for some open $U,V \subset \mathbb{R}$. But we know that the product structure on the plane is the same as the standard structure on the plane, and the standard smooth structure has charts with more complicated coordinate domains then open rectangles $\endgroup$ – joeb Feb 12 '17 at 22:01

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