Q: Place a unit sphere in the $xy$-plane centered at the origin; then draw a line through the North pole $N$ and some point $(x,y,z)$ on the sphere. The line will also cross the $xy$-plane at $(X,Y).$ We thus see that every point (except the North pole) can be mapped to point in $\mathbb R^2.$
Find the mappings between $(x,y,z)$ and $(X,Y).$
I'm not understanding how the formatting of the mappings should be. I am going to use two of these maps to make a smooth structure, create a topology, and then argue it is Hausdorff, but I'm not connecting the dots on what this should look like. A drawing? A set?