While reading the Wikipedia page for covering spaces, I came across a section discussing the gimbal lock issue of Euler angles from the perspective of topology and covering spaces. Shortly, it mentions that Euler angles are defined on a three-torus, while the rotation group $\mathrm{SO}(3)$ is topologically the three-dimensional real-projective space, which has as its "only (non-trivial) covering space the hypersphere $S^3$, which is the group $\mathrm{Spin}(3)$, and represented by the unit quaternions.".
I was wondering whether there are any books or papers that discuss this in more detail. That is, I'm looking for a reference that discuss these topological properties of $\mathrm{SO}(3)$ and their relation to the gimbal lock problem. If that is not possible, references discussing the statement "The only (non-trivial) covering space of $\mathbb{R}P^3$ is the hypersphere $S^3$" would also be very useful.
I'm a physicist. I have an okay understanding of general topology and know very little about covering spaces (in case the question is oddly trivial, that might be the reason). Hence, if possible I'd prefer references that are at a sort of introductory level. If that is not possible, advanced references work as well.
Thanks in advance!