Homeomorphism from the interior of a unit disk to the punctured unit sphere I need help constructing a homeomorphism from the interior of the unit disk,  $\{(x,y)\mid x^2+y^2<1\}$, to the punctured unit sphere, $\{(x,y,z)\mid x^2+y^2+z^2 = 1\} - \{(0,0,1)\}$. I was thinking you could take a line passing through $(0,0,1)$ and a point in the disk and send that point to the part of the sphere the line passes through, but this function wouldn't cover the top half of the sphere.

You can make that homeomorphism by coposing two homeomorphisms $T_1$ and $T_2$, which $T_1:S^3\setminus\{(0,0,1)\}\rightarrow\mathbb{R}^2$ is the  stereographic projection given by:$$T_1(x,y,z) = (\frac{x}{1-z},\frac{y}{1-z})$$
and $T_2$ is any homeomorphism between $\mathbb{R}^2$ and unit ball, $B^2$, writing in the polar coordinate for example:$$T_2(r,\theta) = (\frac{r}{1+r}, \theta)$$
and note that composition of two homeomorphism is a homeomorphism.
