# Homeomorphism between $\mathbb{R^2}$ and $S^2-N$, the sphere without its north pole

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

• See the Riemann sphere and stereographic projection: en.wikipedia.org/wiki/Riemann_sphere – Marc May 24 '16 at 15:47
• The sphere is in $\mathbb{R^3}$ right? But by removing a point it can be deformed into $\mathbb{R^2}$ - is this correct? – thinker May 24 '16 at 16:26
• yes, that's correct. – Thomas May 24 '16 at 16:27
• You are dealing with the surface of the sphere. It might be simpler to visualise if you consider $\mathbb{R}$ and $S- \{(0,1)\}$ first, then extend. – copper.hat May 24 '16 at 16:27
• @Marc I am unsure how this is derived, what is the intuition behind stereographic projection in this case (in a topological sense)? – thinker May 24 '16 at 16:29