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
π1(h(
u)) = ƒ|
ƒ-1(U)(
u) for all
u in ƒ
-1(
U). Here
π1 is the projection
map onto the first coordinate of the
product space U ×
F, i.e.,
π1(
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 S1, which is given by projecting the first coordinate of any point of the Möbius strip onto S1 and the fiber is the unit interval [0, 1].
The similar map from the unit sphere S3 in R4 (or C2 if you like) to S2, whose fiber is S1. If you factorize S3 with respect to the fibers, which may be viewed as equivalence classes, the resulting quotient space is homeomorphic to the complex projective line.