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I'm reading a book and a definition on covering manifold made me thought of fiber bundle.

Wiki's The definition of covering Space and Fiber bundle. It looks to be that they both involved projection, $\pi$, and somewhat decomposition of local space. But somehow I felt they were not quite the same.

Could you explain to me what's the common part and differences between covering manifold and fiber bundle, please?

ShoutOutAndCalculate
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    Fiber bundle is a more general concept. A covering is the same thing as a fiber bundle with discrete fibers. – Moishe Kohan Oct 23 '19 at 22:03
  • @MoisheKohan yeah, I read some thing like 2 sheet covering as an example, but there's no mention of weather the covering could be made into continues or not,, while fiber had to be(not sure) in product space? – ShoutOutAndCalculate Oct 23 '19 at 22:06
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    I do not understand what you are saying. My suggestion is to write down carefully the definition of a fiber bundle in the case when the fiber has discrete topology and compare it to the definition of a covering map. (I forgot to add that the base should be connected, otherwise, a covering map can have fibers of different cardinality.) – Moishe Kohan Oct 23 '19 at 22:10
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    https://math.stackexchange.com/questions/133519/covering-space-is-a-fiber-bundle?noredirect=1&lq=1 – Moishe Kohan Oct 23 '19 at 22:22
  • @MoisheKohan Just made a physical copy of a fiber bundle. I think I got it. Thank you. – ShoutOutAndCalculate Oct 23 '19 at 23:02

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As Moishe Kohan explained in his comments, covering projections are fiber bundles with discrete fibers, at least if the base space $B$ is connected (If it is not, we can write $B$ as the union of nonempty disjoint open subspaces $U_1, U_2$ and there exist coverings with fibers of different cardinality over $U_1, U_2$. This would no longer be a fiber bundle because there is no common fiber.)

On the other hand, coverings have a number of very special features like unique path lifting which is not true for general fiber bundles.

Paul Frost
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