2.NounAn abstract object in topology where copies of one object are "attached" to every point of another, as hairs or fibers are attached to a hairbrush. Formally, a topological space E (called the total space), together with a topological space B (called the base space), a topological space F (called the fiber), and surjective map
π
{\displaystyle \pi }
from E to B (called the projection or submersion), such that every point of B has a neighborhood U with
π
−
1
(
U
)
{\displaystyle \pi ^{-1}(U)}
homeomorphic to the product space U
×
{\displaystyle \times }
F (that is, E looks locally the same as the product space B
×
{\displaystyle \times }
F, although its global structure may be quite different)."A Möbius strip is a fiber bundle which looks locally (i.e., over a connected proper subset of its base space) like the corresponding part of a cylinder
S
1
×
[
0
,
1
]
{\displaystyle S^{1}\times [0,1]}
(a Möbius strip and a cylinder have isomorphic base spaces). A Klein bottle is a fiber bundle which looks locally like the corresponding part of a torus
S
1
×
S
1
{\displaystyle S^{1}\times S^{1}}
(again they could be thought of as sharing the same base space
S
1
{\displaystyle S^{1}}
; cutting out even a single point of that base space makes the cut Klein bottle isomorphic to the cut torus).; In general, a fiber bundle consists of a set of mutually disjoint fibers “over” a base space, which indexes the fibers; there is a copy of some fiber on top of, or projecting (“canonically”) onto each point of the base space."