Im working on a problem about 1 point compactification, and i am at a step where I want to take the punctured sphere $$ S^{2}\setminus\left\{ (0,0,1)\right\}= \left \{ (x,y,z)\in\mathbb{R}^3 \, \mid \, x^2 + y^2 + z^2 = 1 \right\} \setminus\left\{ (0,0,1)\right\} $$
and identify the poles with one another to get the horn torus of radius 1/2 with no centre, parameterized by the following equations (wikipedia for reference): $$ (x,y,z) = \frac{1}{2} \left( (1+\cos\theta ) \cos\varphi, (1+\cos\theta ) \sin\varphi, \sin\theta \right) $$
The question does not require me to state the homeomorphism explicity, but I have been trying for too long and at this point need to know for the sake of interest...
I have tried various ways of mapping spherical coordinates into this torus, but i think that is the wrong approach since it seems to always have a dead end
Map that sends points from the torus to the sphere st. point is sent to it's projection through the origin works (cf. attached picture)