Collapsing the unit sphere of the unit ball $S^{n-1} \subset D^n \subset \mathbb{R}^n$ to a point. How can I show that it ($D^n/S^{n-1}$) is homeomorphic to the sphere of one dimension higher ($S^n$)?
I thought about $D_n$ = {($x_1$, ..., $x_n$): $x_1$+$x_n$$\leqslant 1$} and $S_{n-1} = ${($x_1$, ..., $x_n$): $x_1$+...+$x_n$ = $1$}. So $D_n / S_{n-1}$ = $\bigcup_{z \in \left[-1, 1\right]}$ {($x_1$, ..., $x_n$, z): ${x_1}^2 + ... + {x_n}^2 = 1 - z^2$} = $\bigcup_{z \in \left[-1, 1\right]}$ {($x_1$, ..., $x_n$, z): ${x_1}^2 + ... + {x_n}^2 + z^2 = 1$}. And it is clearly similar to $S_{n}$. Can we use my idea to construct this homeomorphism sending ($x_1$, ..., $x_n$) in ($x_1$, ..., $x_n$, $z$)?