Undergrad Is the Boundary Chart for a Closed Unit Ball Injective?

[JYM]

I want to show that a closed unit ball is manifold with boundary and I attempted as uploaded. But I am not happy with the way I showed the boundary chart is injective. Am I right?

 

[andrewkirk]

At the bottom of p1, the assumption that $U^+$ lies in the closed upper half-ball will not hold if the point $p$ has zero $i$th coordinate.

It looks like you are defining a global map $\phi$ and trying to show its restriction to a nbd of a boundary point on the hypersphere is a homeomorphism to a half-ball. I don't think that will work because of equator points wrt the $i$th dimension. Instead, choose the putative homeomorphism based on the location of the boundary point. A boundary point must have at least one nonzero coordinate. Say the first nonzero coord of the point is the $i$th coordinate, then define $\pi_i$ to be the projection map that removes the $i$th coord.

That should enable you to get around the obstacle.

 

[JYM]

I see it. Thanks!