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A fibre bundle (cf. [[Fibre space|Fibre space]])  $  \pi :  X \rightarrow B $
with fibre  $  F $
such that for any point of the base  $  b \in B $
there is a neighbourhood  $  U \ni b $
and a homeomorphism  $  \phi _ {U} :  U \times F \rightarrow \pi  ^ {-} 1 ( U) $
such that  $  \pi \phi _ {U} ( u, f  ) = u $, 
where  $  u \in U $, 
$  f \in F $. 
The mapping  $  h _ {U} = \phi _ {U}  ^ {-} 1 $
is called a chart of the locally trivial bundle. The totality of charts  $  \{ h _ {U} \} $
associated with a covering of the base  $  \{ U \} $
forms the atlas of the locally trivial bundle. For example, a [[Principal fibre bundle|principal fibre bundle]] with a locally compact space and a Lie group  $  G $
is a locally trivial fibre bundle, and any chart  $  h _ {U} $
satisfies the relation

$$ 
h _ {U} ( g x )  =  g h _ {U} ( x) ,\ \ 
x \in \pi  ^ {-} 1 ( U) ,
$$

where  $  G $
acts on  $  G \times U $
according to the formula  $  g ( g  ^  \prime  , u ) = ( g g  ^  \prime  , u ) $. 
For any locally trivial fibre bundle  $  \pi :  X \rightarrow B $
and continuous mapping  $  f :  B _ {1} \rightarrow B $
the [[Induced fibre bundle|induced fibre bundle]] is locally trivial.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Steenrod,   "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.-T. Hu,   "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>