As someone who trained as a physicist, I have known for ages that $\operatorname{SU}(2)$ is a double cover of $\operatorname{SO}(3)$, so, during an idle day at the office I decided to look up what this meant. It turned out to be more complicated than I had expected. The definition of a covering space of a topological space seemed to be quite fiddly and left me thinking that these things were born of utility rather than essential beauty.
So, what are they for, and why are they more useful, than say a simple open cover? And why are the pre-images of a point in the covered space called a ‘fibre’ (is it linked to fibre bundles in differential geometry?). And, all of the many definitions I have seen seem to imply that the fibres must be discrete (presumably, countably infinite at the most, does this follow from the definitions?), and beyond that they always seem to be finite and of the same number (i.e., always $2$ at every point in the case of $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$) is this always the case?