Editing Associated bundle (section)

===Associated bundles in general===
Let <math display="inline">\pi:E\to X</math> be a fiber bundle over a [[topological space]] <math>X</math> with structure group <math>G</math> and typical fibre <math>F</math>.  By definition, there is a [[Group action (mathematics)|left action]] of <math>G</math> (as a [[transformation group]]) on the fibre <math>F</math>.  Suppose furthermore that this action is [[Group action (mathematics)#Notable properties of actions|effective]].<ref>Effectiveness is a common requirement for fibre bundles; see Steenrod (1951).  In particular, this condition is necessary to ensure the existence and uniqueness of the principal bundle associated with <math>E</math>.</ref>
There is a [[locally trivial|local trivialization]] of the bundle <math>E</math> consisting of an [[open cover]] <math>U_i</math> of <math>X</math>, and a collection of [[bundle map|fibre maps]]<math display="block">\varphi_i : \pi^{-1}(U_i) \to U_i \times F</math>such that the [[transition map]]s are given by elements of <math>G</math>.  More precisely, there are continuous functions <math>g_{ij} \colon U_i \cap U_j \to G</math> such that<math display="block">\psi_{ij}(u,f) := \varphi_i \circ \varphi_j ^{-1}(u,f) = \big(u, g_{ij}(u) f \big),\quad 
\text{for each } (u,f)\in (U_i \cap U_j)\times F\, .</math> 

Now let <math>F'</math> be a specified topological space, equipped with a continuous left action of <math>G</math>. Then the bundle '''associated''' with <math>E</math> with fibre <math>F'</math> is a bundle <math>E'</math> with a local trivialization subordinate to the cover <math>U_i</math> whose transition functions are given by<math display="block">\psi'_{ij}(u,f') = \big(u, g_{ij}(u) f' \big),\quad 
\text{for each } (u,f')\in (U_i \cap U_j)\times F'\,,</math>where the <math>G</math>-valued functions <math>g_{ij}(u)</math> are the same as those obtained from the local trivialization of the original bundle <math>E</math>. 
This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of <math>G</math>-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the <math>g_{ij}</math> transform via the same coboundary.) Hence, by the [[fiber bundle construction theorem]], this produces a fibre bundle <math>E'</math> with fibre <math>F'</math> as claimed.