Local trivialization

Summary

A local trivialization of a map $\pi \colon E \to B$ over an open set $U$ is a homeomorphism $\pi^{-1}(U) \to U \times F$ over $U$. It says that over $U$ the map $\pi$ looks exactly like the projection of a product. A fiber bundle is a map that has a local trivialization around every point of the base. Comparing two local trivializations on an overlap gives the transition functions, which encode how the bundle twists.