we defined the projective space as $\mathbb{S^2/Z_2}$ i.e. identify antipodal points and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$
And now I am concerned with their manifold structure-
In fact, I managed to cover both of them with three charts and it is also immediate to me that one chart cannot cover them, because they are compact, so their image would be compact under a homeomorphism. Despite, I don't see why two charts cannot cover them. Is there an easy argument for this?