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A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.

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Are all quotients of a weakly contractible space via a free group action classifying spaces ...

I asked this question on math.stackexchange a week ago, but did not get an answer. First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situ …
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