Editing Fiber bundle (section)

=== Nontrivial bundles ===

==== Möbius strip ====

[[File:Moebius Surface 1 Display Small.png|thumb|right|The Möbius strip is a nontrivial bundle over the circle.]]
Perhaps the simplest example of a nontrivial bundle <math>E</math> is the [[Möbius strip]]. It has the [[circle]] that runs lengthwise along the center of the strip as a base <math>B</math> and a [[line segment]] for the fiber <math>F</math>, so the Möbius strip is a bundle of the line segment over the circle. A [[neighbourhood (mathematics)|neighborhood]] <math>U</math> of <math>\pi(x) \in B</math> (where <math>x \in E</math>) is an [[circular arc|arc]]; in the picture, this is the [[length]] of one of the squares. The [[image (mathematics)|preimage]] <math>\pi^{-1}(U)</math> in the picture is a (somewhat twisted) slice of the strip four squares wide and one long (i.e. all the points that project to <math>U</math>).

A homeomorphism (<math>\varphi</math> in {{sectionlink||Formal definition}}) exists that maps the preimage of <math>U</math> (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle <math>B\times F</math> would be a [[cylinder (geometry)|cylinder]], but the Möbius strip has an overall "twist". This twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).

==== Klein bottle ====

A similar nontrivial bundle is the [[Klein bottle]], which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-[[torus]], <math>S^1 \times S^1</math>.

{|
|[[File:KleinBottle-01.png|thumb|120px|The Klein bottle [[Immersion (mathematics)|immersed]] in three-dimensional space.]]
|[[File:Torus.png|thumb|160px|A torus.]]
|}