How many charts are needed to cover a 2-torus?

If the images of the charts are not restricted to be simply connected (which they are not according to the Wikipedia definition of "chart"), then there is a two-chart-system covering the torus.

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And even with simply connected open sets, there is a way to do it with only three charts. Here is a picture showing how these charts look like on the flat torus (the right one shows that they indeed cover everything).

Four will do it. Wrap one around the outside of the doughnut and another on the inside. Let them overlap a little. Both of these are diffeomorphic to hollow cylinders, which require two patches to cover.

Hint $\mathbb T^2=\mathbb S^1\times\mathbb S^1$

It is easily to see that $\mathbb S^1$ can be charted by two covers, then $\mathbb T^2$ is four.