Atlas of Manifold: Is it Countable?

[Pietjuh]

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 :)

 

[George Jones]

Each n-sphere S^n is covered by an atlas that has 2 members.

Do you mean maximal atlas?

Regards,
George

 

[homology]

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