Local trivialization
= Local trivialization
A local trivialization of a <fiber bundle> $p:E\to B$ over an open set $U\subseteq B$ is a <homeomorphism> $p^{-1}(U)\cong U\times F$ commuting with the projections to $U$. For a <vector bundle>, it must be linear on each fiber. Compatible local trivializations specify the bundle structure and give its transition functions on overlaps.