 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.