A

fiber bundle ƒ:

*E* →

*B* with

fiber *F* is a

map such that every point

*b* of the base

space *B* has a

neighborhood *U* such that there exists a

homeomorphism *h*: ƒ

^{-1}(

*U*) →

*U* ×

*F*, where

*π*_{1}(h(

*u*)) = ƒ|

_{ƒ-1(U)}(

*u*) for all

*u* in ƒ

^{-1}(

*U*). Here

*π*_{1} is the projection

map onto the first coordinate of the

product space *U* ×

*F*, i.e.,

*π*_{1}(

*u*,

*f*) =

*u* for all (

*u*,

*f*) in

*U* ×

*F*, and ƒ|

_{ƒ-1(U)} is the function ƒ with

domain resticted to ƒ

^{-1}(

*U*).

What this means is that the total space *E* of the fiber bundle locally resembles the product space *B* × *F*, i.e., points in *E* have neighborhoods homeomorphic to neighborhoods of the product space *B* × *F*. The total space *E* is much like a manifold and this is, in fact, an important area in which fiber bundles are studied and used.

Since the fibers ƒ^{-1}(*b*) for *b* in the base space *B* are isomorphic in whatever category is being used the particular choice of ƒ^{-1}(*b*) as representative of *F* is irrelevant.

Particular types of fiber bundles are vector bundles and principle bundles.

Examples of fiber bundles are:

Covering maps, which are precisely those fiber bundles whose fibers are discrete, i.e., the topology of the fibers with respect to the total space *E* is the discrete topology

The map from the Möbius strip to the unit circle *S*^{1}, which is given by projecting the first coordinate of any point of the Möbius strip onto *S*^{1} and the fiber is the unit interval [0, 1].

The similar map from the unit sphere *S*^{3} in **R**^{4} (or **C**^{2} if you like) to *S*^{2}, whose fiber is *S*^{1}. If you factorize *S*^{3} with respect to the fibers, which may be viewed as equivalence classes, the resulting quotient space is homeomorphic to the complex projective line.