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.

Stereographic projection, as mentioned in the other answers, usually comes up in this context. Indeed, it is a beautiful way of identifying the punctured sphere with $\mathbb{R}^2$, in a conformal manner (preserving angles).
However, if you only care about finding a homeomorphism to the disk (and there is no way to make this conformal anyway),  then perhaps it is easier to just define the homeomorphism from the punctured sphere to the unit disk by
$$ \phi(x,y,t) := \frac{t+1}{2} e^{i\theta}, $$
where $\theta$ is the angle of the point in the $x-y$-plane; i.e. $\theta=\arg(x+iy)$.
If you prefer real coordinates, this means
$$ \phi(x,y,t) = \frac{t+1}{2\sqrt{x^2+y^2}} \cdot (x,y).$$
