B How to select the smooth atlas to use for spacetime?

[cianfa72]

He wasn't being careful. He probably meant that the two are the same topological space with two different smooth structures. But in dimension 2 the to different structures are diffeomorphic.

Watching it again, I believe as follows:

• for both the regular and the sphere with 'edge' he takes the subspace topology inherited from $\mathbb R^3$
• for the regular sphere all the smooth charts in $\mathbb R^3$ 'restricted' to the sphere are compatible and thus define a maximal atlas for it
• however for the sphere with 'edge' we cannot take all those charts because when 'restricted' to it are not longer mutually compatible

Thus the sphere with 'edge' and the regular one have not the same smooth structure (actually the regular sphere is a submanifold of $\mathbb R^3$ whereas the other is not) nevertheless are diffeomorphic as manifolds of dimension 2