 Fiber bundle

A fiber bundle consists of a continuous map $\pi :E\to B$ and a model fiber $F$ such that every point $b\in B$ has an open neighborhood $U$ and a local trivializationThus a nearby family of fibers fits together as a product, while changes between local trivializations can twist those products globally. In the smooth category, $E,B,F$ are smooth manifolds and the local trivializations are diffeomorphisms.

The Möbius line bundle has real-line fibers over a circle. Cutting the circle along a small arc trivializes it, but going once around the circle reverses a fiber coordinate. It is a fiber bundle without being the trivial vector bundle. Local triviality requires a suitable neighborhood of each point; it does not assert a product over every open subset of the base.