How would one approach the following problem?
Write down a homeomorphism and its inverse from $\mathbb{R^2}$ to the sphere $S^2-N$ without its north pole
So I need a function $f(x,y) : \mathbb{R^2} \rightarrow S^2-N$
$\mathbb{R^2}=\{(x, y) | x, y \in \mathbb{R}\}$
$S^2=\{(x, y)\in\mathbb{R^2}|d(x, y)=r\}$ where $r$ is the radius of the sphere
$S^2-N=\{(x, y)\in\mathbb{R^2}|d(x, y)=r, (x, y)\neq N\}$
I think $f$ could be some retraction but I am unsure how to formulate it