Real projective plane ($R{P}^2$, $\mathbf{P}({\mathbb{R}}^3)$)

For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.

To see why this is called a plane, move he center of the sphere to $z=1$, and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator $z=1$. Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.

It good to think about how Euclid's postulates look like in the real projective plane:

Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.

The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.

One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.

This is the standard model.

Ciro Santilli's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in ${\mathbb{R}}^3$".

Take a open half sphere e.g. a sphere but only the points with $z>0$.

Each point in the half sphere identifies a unique line through the origin.

Then, the only lines missing are the lines in the x-y plane itself.

For those sphere points in the circle on the x-y plane, you should think of them as magic poins that are identified with the corresponding antipodal point, also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.

Ciro likes this model because then all the magic is confined just to the $z=0$ part of the model, and everything else looks exactly like the sphere.

It is useful to contrast this with the sphere itself. In the sphere, all points in the circle $z=0$ are the same point. But this is not the case for the projective plane. You cannot instantly go to any other point on the $z=0$ by just moving a little bit, you have to walk around that circle.

To see that the real projective plane is not simply connected space, considering the lines through origin model of the real projective plane, take a loop that starts at $(1,0,0)$ and moves along the $y=0$ great circle ends at $(-1,0,0)$.

Note that both of those points are the same, so we have a loop.

Now try to shrink it to a point.

There's just no way!