So $T^2$ double covers $S^2$, and $S^2$ double covers $\mathbb{R}P^2$. Therefore, $T^2$ quadruple covers $\mathbb{R}P^2$.
I am looking at $\mathbb{R}P^3$ because I am interested in rotations. The Euler angles, for example parametrize $\mathbb{R}P^3$ once you make a restriction on one of the angle's ranges. Originally though, they parametrize $T^3$. Judging from the restriction of the angle, I have come to the conclusion that $T^3$ double covers $\mathbb{R}P^3$. Is this true? And how else would I prove it?
Thanks
There is a map from $T^3$ to $SO(3)$, called "Euler angles" -- basically, you use each angular coordinate to specify "yaw", "pitch", and "roll". This map is, however, singular, much like the map from $T^2$ to $S^2$ discussed in the comments. The singularities are often referred to by the generic name "gimbal lock". You can read about this in books on robotics, or in Wikipedia, or in some computer graphics books.
Note that the euler-angle parametrization, because of its singularities, is not a covering in the sense of covering-spaces, nor does it have many other properties you might hope for, like some sort of equivariance. It's a totally great parameterization near the identity, and gets worse and worse as you get farther from $I$.
What order do pitch, roll, and yaw come in? There are six possible orderings, and I'm pretty sure all have bene used at one time or another. There's no firm consensus.
This particular kind of parameterization of $SO(3)$ turns out to be not very useful for a lot of computer graphics applications, nor for many applications in mathematics. For control of "flying things" in games, it has some value, as it mimics airplane controls (somewhat).