Fiber bundle

Remarks

For any spaces $B$ and $F$, the product $B \times F$ with $\pi = \operatorname{pr}_1$ is a fiber bundle, the trivial bundle; one trivialization over $U = B$ suffices. The Möbius band is a fiber bundle over the circle with fiber $[0, 1]$ that is not trivial.

Non-example: let $E = \set{(x, y) \in \R^2 : xy = 0}$ and $\pi(x, y) = x$, so $\pi \colon E \to \R$. The preimage of $0$ is a line, while the preimage of each $x \neq 0$ is a single point. A local trivialization around $0$ would make all preimages near $0$ homeomorphic to the same $F$, which is impossible.

Each fiber $\pi^{-1}(b)$ is homeomorphic to $F$. A smooth fiber bundle uses smooth manifolds, a smooth submersion $\pi$ and diffeomorphisms $\varphi$. The notation $F \to E \to B$ is common.