Show the intersection of two parts of an $n$-sphere, each homeomorphic an $n$-disk have a union that's homeomorphic to an $n-1$-sphere.

Question

I really wasn't sure where to begin with this. Any help would be appreciated.

Thanks in advance!

$S^n=\{x \in R^{n+1}, ||x||=1\}$ and $D^n=\{x \in R^n, ||x||\le1\}$

Answer 1

For our earth we know such a union: northern hemisphere, southern hemisphere; their intersection is the equator..

Formally:

$U = \{x \in S^n: x_{n+1} \ge 0\}$ is upper,

$L= \{x \in S^n: x_{n+1} \le 0\}$ is lower.

Both are closed as e.g. $U=S^n \cap \pi_{n+1}^{-1}[[0,\infty)]$ is the intersection of closed sets etc.

$ U \cap L = \{x \in S^n: x_{n+1}=0\}$ which is obviously homeomorphic to $S^{n-1}$ (just ditch the last coordinate and we have a homeomorphism). And $U$ and $L$ are both homeomorphic to $D^n$ by the map $h(x_1,\ldots,x_n, x_{n+1})= (x_1,\ldots,x_n)$. The inverse for $U$ is $(x_1, x_2, ,\ldots, x_n) \to (x_1, \ldots, x_n, \sqrt{1-\sum_{i=1}^n x_i^2})$, while for $L$ we negate the final coordinate of that map.