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Definition/Summary

Intuitively speaking, a fibre bundle is space E which ‘locally looks like’ a product space B×F, but globally may have a different topological structure.

Extended explanation

Definition:

A fibre bundle is the data group [itex](E, B,\pi, F)[/itex], where [itex]E, B[/itex], and [itex]F[/itex] are topological spaces called the total space, the base space, and the fibre space, respectively and [itex]\pi : E \rightarrow B[/itex] is a continuous surjection, called the projection, or submersion of the bundle, satisfying the local triviality condition.

(We assume the base space B to be connected.)

The local triviality condition states the following:

we require that for any [itex]x \in E[/itex] that there exist an open neighborhood, [itex]U[/itex] of [itex]\pi (x)[/itex] such that [itex]\pi^-1[/itex][itex](x)[/itex] is homeomorphic to the product space [itex]U×F[/itex] in such a manner as to have [itex]\pi[/itex] carry over to the first factor space of the product...

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