locally trivial bundle

A locally trivial bundle is a continuous map$\pi :E\to B$ of topological spaces such that the following conditionshold.First, each point $x\in B$ must have a neighborhood $U$ such thatthe inverse image $\tilde{U}={\pi }^{-1}(U)$ is homeomorphicto $U\times {\pi }^{-1}(x)$.Second, for some homeomorphism $g:\tilde{U}\to U\times {\pi }^{-1}(x)$,the diagram

must be commutative (http://planetmath.org/CommutativeDiagram).

Locally trivial bundles are useful because of their covering homotopy property and because each locally trivial bundle has an associatedlong exact sequence (http://planetmath.org/LongExactSequenceLocallyTrivialBundle) and Serre spectral sequence. Every fibre bundle is an example of a locally trivial bundle.