# Is there anywhere we use a fibration which is not a fiber bundle

What I currently meet are all fiber bundles.

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–  BBischof Oct 11 '10 at 4:26

There's a canonical example of fibrations used in algebraic topology which are not fibre bundles. That is, given a continuous function $f : X \to Y$ there is a homotopy-equivalence $\phi : X' \to X$ and a fibration $f' : X' \to Y$ such that $f \circ \phi$ is homotopic to $f'$. I believe this idea goes back to Serre (or perhaps earlier). The fibre of $f'$ is called the homotopy-fibre of $f$.
A common usage of this construction is with the Postnikov Tower of a space. In his dissertation Jean-Pierre Serre used this (and some "closure" observations of the Serre spectral sequence of a fibration, then it was called Serre $\mathcal C$-theory, nowadays this technology is subsumed in localization) to show that most of the homotopy-groups of the spheres are finite.