The sphere $S^n$ with the requirement that $x_{n+1} \geq 0$ is homeomorphic(topological equivalent) to the ball $B^n$. I mean the euclidean sphere with the euclidean metric, even if that makes a difference.
So I just wondering, is it the case that if you cut a small piece out of sphere i.e. take 1 cut point in the sphere is that the same ball.
As I imagine if you do take a cut point. The sphere would have a hole in it? why can't you then stretch that hole such that it looks like the top hemisphere of a sphere.