Editing Fiber bundle (section)

== Formal definition ==

A fiber bundle is a structure <math>(E,\, B,\, \pi,\, F),</math> where <math>E, B,</math> and <math>F</math> are [[topological spaces]] and <math>\pi : E \to B</math> is a [[Continuous (topology)|continuous]] [[surjection]] satisfying a ''local triviality'' condition outlined below. The space <math>B</math> is called the '''{{visible anchor|base space}}''' of the bundle, <math>E</math> the '''{{visible anchor|total space}}''', and <math>F</math> the '''{{visible anchor|fiber}}'''. The map <math>\pi</math> is called the '''{{visible anchor|projection map}}''' (or '''{{visible anchor|bundle projection}}'''). We shall assume in what follows that the base space <math>B</math> is [[Connected space|connected]].

We require that for every <math>x \in B</math>, there is an open [[Neighborhood (topology)|neighborhood]] <math>U \subseteq B</math> of <math>x</math> (which will be called a trivializing neighborhood) such that there is a [[homeomorphism]] <math>\varphi : \pi^{-1}(U) \to U \times F</math> (where <math>\pi^{-1}(U)</math> is given the [[subspace topology]], and <math>U \times F</math> is the product space) in such a way that <math>\pi</math> agrees with the projection onto the first factor. That is, the following diagram should [[Commutative diagram|commute]]:

[[image:Fibre bundle local trivial.svg|Local triviality condition|230px|center]]

where <math>\operatorname{proj}_1 : U \times F \to U</math> is the natural projection and <math>\varphi : \pi^{-1}(U) \to U \times F</math> is a homeomorphism. The [[Set (mathematics)|set]] of all <math>\left\{\left(U_i,\, \varphi_i\right)\right\}</math> is called a '''{{visible anchor|local trivialization}}''' of the bundle.

Thus for any <math>p \in B</math>, the [[preimage]] <math>\pi^{-1}(\{p\})</math> is homeomorphic to <math>F</math> (since this is true of <math>\operatorname{proj}_1^{-1}(\{p\})</math>) and is called the '''fiber over <math>p.</math>''' Every fiber bundle <math>\pi : E \to B</math> is an [[open map]], since projections of products are open maps. Therefore <math>B</math> carries the [[quotient topology]] determined by the map <math>\pi.</math>

A fiber bundle <math>(E,\, B,\, \pi,\, F)</math> is often denoted
{{NumBlk
|:| <math>\begin{matrix}
  {} \\
  F \longrightarrow E\ \xrightarrow{\,\ \pi\ }\ B \\
  {}
\end{matrix}</math>
 | {{EquationRef|1}}
}} that, in analogy with a [[short exact sequence]], indicates which space is the fiber, total space and base space, as well as the map from total to base space.

A '''{{visible anchor|smooth fiber bundle}}''' is a fiber bundle in the [[Category (mathematics)|category]] of [[smooth manifold]]s. That is, <math>E, B,</math> and <math>F</math> are required to be smooth manifolds and all the [[Function (mathematics)|functions]] above are required to be [[smooth map]]s.