Saying it more formally, for any x in B, there is a neighborhood such that is homeomorphic to , in such a way that π carries over to the projection onto the first factor.

B is called the **base space** of the fiber bundle and for any , the preimage of x, is called the fiber at x and the map π is called the projection map.

A standard example is a Möbius strip as E, in which B can be taken as a circle and F a line segment. The 'twisting' in the band is only apparent globally, while locally the ribbon structure defines the topology.

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Sometimes, there exists a topological group G of transformations of E, such that if ρ denotes the action,

In that case, G is called the structure group of the fiber bundle. To qualify as a G-bundle, the matching conditions between local trivializable neighborhoods would have to be intertwiners of G-actions as well.

If, in addition, G acts freely, transitively and continuously upon each fiber, then we call the fiber bundle a **principal bundle**. An example of a principal bundle that occurs naturally in geometry is the bundle of all bases for the tangent space to a manifold, with G the general linear group; restricting in Riemannian geometry to orthonormal bases, one would limit G to the orthogonal group. See vierbein for more details.

Making G explicit is essential for the operations of creating an associated bundle, and making precise the reduction of the structure group of a bundle.

A section of a fiber bundle is a continuous map, such that , for x in B. Since bundles do not in general have sections, one of the purposes of the theory is to account for their existence. This leads to the theory of characteristic classes in algebraic topology.

One of the primary applications of fiber bundles is in gauge theory.

See also Fibration.

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