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Here is where I read "trivializing open cover", but I was not able to find out what it is.

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Eric O. Korman
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WishingFish
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    I believe a trivializing open cover is an open cover of M by such neighborhoods U as in the first paragraph. – Dylan Yott Jun 24 '13 at 14:54

1 Answers1

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It's just as Dylan Yott says in his comment. A fiber bundle is by definition locally trivial: around every point there is a neighborhood where the bundle is isomorphic [in whatever category; here, differentiably] to a "trivial bundle", i.e., just taking the product of the base with the fiber. So to define the bundle using transition functions as is being done here, one needs to cover the base by open neighborhoods $U_i$ such that the bundle is trivial -- i.e., isomorphic to a product -- on each $U_i$. That's a trivializing open cover.

Pete L. Clark
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