View source for Locally trivial fibre bundle

A fibre bundle (cf. [[Fibre space|Fibre space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605301.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605302.png" /> such that for any point of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605303.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605304.png" /> and a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605305.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605308.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605309.png" /> is called a chart of the locally trivial bundle. The totality of charts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053010.png" /> associated with a covering of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053012.png" /> is a locally trivial fibre bundle, and any chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053013.png" /> satisfies the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053015.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053016.png" /> according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053017.png" />. For any locally trivial fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053018.png" /> and continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053019.png" /> 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>