Topological properties of the so-called "plane at infinity". When 3D-Euclidean geometry is extended with ideal points at infinity, a whole "plane at infinity" is added to the geometry.
Apart from metric properties it has become a 3D projective space and 
the plane at infinity is a regular projective plane.   
On the other hand we experience infinity as a "sphere" around us at infinite distance.
I mean in the real world.
Does anybody have suggestions on how to bring the view of a plane and a sphere together, looking at topological properties of the "plane at infinity"?
Put in other words: what kind of topological model for the "plane at infinity" would be appropiate?

In $\mathbb R^4,$ take the ordinary 3-sphere
$$ x^2 + y^2 + z^2 + w^2 = 1.  $$
For anything on this, centrally project from $(0,0,0,0)$ to the hyperplane
$$ w = -1.  $$
The points that are pushed out to infinity are the equator of the 3-sphere, which is a 2-sphere, but with antipodal points identified, so it is an RP2 
