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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?

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    Dear James, The definition only seems fiddly if you don't have experience and a broader context in which to place it. (Just as the definition of group, or normal subgroup,2011-12-31
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    I'm certain that that's true. Any chance of an example of this broader context? I am not scared of the fiddly, but then again I prefer elegant, so I hope there was some great need for this. After all, to define the Lesbesgue integral you could take the Caratheodary route or the representation theory route - both fiddly - but at least you know why you were putting the effort in. What wrong with a simple open cover? And why does the 'fiber' make me think of fiber bundles?2011-12-31
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    @James: Because a covering space is (locally) a fibre bundle! (The fibres are discrete, however.)2011-12-31
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    Ok, so my intuition of link to diff. geom. was not misplaced. However, you say the fibres are discrete (I guess this is in the topological sense, so my countability comment is discounted - but still, is that a requirement or just a consequence of the basic recipe?). Also, my naive notion of a fibre bundle is attaching the same thing to each point of some base manifold. In this case, how do we know that the fibres have the same size - or perhaps they don't? (Apparently some people allow empty fibres, but then what is the point of all this - which is my question!)2011-12-31
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    Actually I bring the countable thing back. These are dis-joint sets in the covering space. Why are they always of the same finite number for all the (err, one..) examples that I care about. And also, again why do people care about these things? They must have evolved se a fertile setting to solve loads of maths problems?2011-12-31
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    Asking why a covering space is more useful than an open cover is comparing apples to oranges; even though both concepts may fall under the label "cover of a space", they are completely different concepts, just like distributions in the sense of probability theory and in the sense of generalized functions.2011-12-31
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    Dear James, Sorry for the slightly fragmented comment. It is now amplified as an answer. Regards,2011-12-31

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