TITLE: Covering spaces - why are they useful?
QUESTION [30 upvotes]: 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?

REPLY [17 votes]: The definition probably only seems fiddly if you haven't seen it (or related)
definitions before.  What is says is the following: a map $p: Y \to X$ is
a covering map if $p$ locally looks like the projection from $$X \times \text{ a discrete space} \to X.$$
A little more precisely: each point $x \in X$ has a neighbourhood $U$ such that the map
$$p^{-1}(U) \to U$$ is isomorphic to a projection
$$U \times \text{ a discrete space} \to U.$$
This kind of property of a map --- that locally on the target it looks like
the projection from a certain kind of product --- is very common in topology and geometry, and underlies the fundamental notion of a fibre bundle.  Covering spaces are perhaps the simplest example, since they are fibre bundles with discrete fibres.  Fibre bundles of all kinds appear everywhere, and so it is not so much a question of asking what they are useful for, but rather, of identifying a ubiquitous property and giving it a name.
Another, more global, way to describe covering spaces is as follows: if a discrete group $\Gamma$ acts on a space $Y$ in such a way that each point has
a neighbourhood $U$ such that the orbits $\gamma U$ are distinct for distinct elements $\gamma \in \Gamma$, then the quotient map $Y \to Y/\Gamma$ is a
covering map (i.e. $Y$ is a covering space of $Y/\Gamma$). 
Since group actions on spaces are pretty ubiquitous, this gives some indiction
of why covering maps might be commonly encountered in topology.  (The basic example
is $\Gamma = \mathbb Z$ acting by translation on $Y = \mathbb R$, with
the cover being $\mathbb R/\mathbb Z$, which is a circle.)
Finally, if you begin with space $X$, in order to construct covers of $X$,
you have to "unwind" certain directions in $X$.  Thus investigating covering
spaces of $X$ is the same as investigating the extent to which the various directions in $X$ are "wound up".  
E.g. in the circle there is just one direction, and unwinding it, you get the covering space $\mathbb R$.  In $SO(3)$ there is one direction which is wound up, and unwinding it gives $SU(2)$.  Often this "unwinding" can be thought of
in a physical way: e.g. imagine that you are walking around a stadium, and measuring the distance you have travelled as you walk.  When you get all the way around, you are back where you are started (the stadium is a circle), but your
distance travelled isn't at zero (it's at 400 metres, say).  The numerical distance travelled "unwinds" the circle of the stadium into the line $\mathbb R$.  
E.g. in $SO(3)$, the "belt trick" mentioned by Georges allows you to "unwind" a rotation to get an element of $SU(2)$.  (And when you do it twice, you get back
where you started --- unlike in the case of the stadium, where your distance travelled never resets to zero; so here you see the difference between a double cover, like $SU(2)$ over $SO(3)$, and an infinite cover, like $\mathbb R$ over $\mathbb Z$.)