I was thinking about something yesterday and I couldn't quite figure it out. It's about the question if an atlas is a countable set. Because we know that every manifold is second countable, so it has a countable basis. But does every element of the basis fit inside a chart domain? If that's the case then the atlas is countable. But I'm not sure that's the case :)
Each n-sphere S^n is covered by an atlas that has 2 members. Do you mean maximal atlas? Regards, George
Not only that but manifolds aren't necessarily second countable. It depends on the author. If you start with a set M, and put a complete smooth atlas on it (so I'm talking about differentiable manifolds in this context), then the charts form a basis for the topology on the M and that topology isn't necessarily second countable. kevin