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?
 A: 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 
A: Think of a family of all lines passing through a point in $\mathbb{R}^3$. Now start moving this point to infinity. In the limit the family turns into a family of parallel lines. By definition, the plane at infinity is the set of these families, it's a projective plane, not an ordinary Euclidean plane. 
Different families have different directions so it seems that we can use the sphere to represent them. However, antipodal directions produce the same family of parallel lines, so we would have to identify antipodal points on the sphere to represent the plane at infinity faithfully. And that's a standard way to describe the projective plane. Many topological properties of the projective plane can be established by using this construction of it as a quotient of the sphere.
A: The plane at infinity is the same as any plane in the projective geometry sense. So it is a projective plane. It is naturally connected to the sphere in two ways:


*

*stereographic projection. Here the line at infinity (of the projective plane) is identified with the north pole of the sphere (suppose that the projective plane is tangent to the sphere at the south pole).

*identifying antipodal points on a sphere with a line passing through the origin. An arbitrary plane not passing through the sphere's origin will meet the line in a unique point. When the line you pick is in parallel with the plane, the intersection point is at infinity.
A: There are lots of ways to compactify $\mathbb{R}^3$ by adding points at infinity; the real projective 3-space $\mathbb{P}^3$ is not the only way to do so.
One way that seems to correspond to your description is the fact that $\mathbb{R}^3$ is homeomorphic to the open unit ball via the homeomorphism in spherical coordinates:
$$ (\rho, \theta, \varphi) \mapsto \left( \frac{\rho}{1+\rho}, \theta, \varphi \right) $$
There isn't anything special about $\rho/(1+\rho)$; I simply chose it beause it's a continuous, increasing function $[0, \infty) \mapsto [0, 1)$.
Since you can compactify the open ball by adding its boundary — the unit sphere $S^2$ — we get a corresponding compactification of $\mathbb{R}^3$, which I will call $X$.
The space $X \setminus \mathbb{R}^3$ consisting of the ideal points at infinity will be homeomorphic to the unit sphere $S^2$.
In terms of a more classical construction, this corresponds to adding one ideal point at infinity for every class of parallel rays in $\mathbb{R}^3$. The construction of the projective space $\mathbb{P}^3$ uses classes of parallel lines instead.
The identity map on $\mathbb{R}^3$ can be extended to a continuous function $X \to \mathbb{P}^3$. On the ideal points at infinity, this map is (equivalent to) the two-fold cover $S^2 \to \mathbb{P}^2$ obtained by identifying antipodal points of the sphere.
