Editing Vector bundle (section)

===Subbundles===
{{Main article|Subbundle}}
[[File:Subbundle.png|thumb|300px|A line subbundle <math>L</math> of a trivial rank 2 vector bundle <math>E</math> over a one-dimensional manifold <math>M</math>.]]
One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle <math>\pi: E\to X</math> over a topological space, a subbundle is simply a [[Linear subspace|subspace]] <math>F\subset E</math> for which the [[Restriction of a map|restriction]] <math>\left.\pi\right|_F</math> of <math>\pi</math> to <math>F</math> gives <math>\left.\pi\right|_F: F \to X</math> the structure of a vector bundle also. In this case the fibre <math>F_x\subset E_x</math> is a vector subspace for every <math>x\in X</math>.

A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, the [[Möbius band]], a non-trivial [[line bundle]] over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.