Fibration

A fibration (or Hurewicz fibration) is a continuous mapping p : E → B satisfying the homotopy lifting property with respect to any space. Fiber bundles (over paracompact bases) constitute important examples. In homotopy theory any mapping is 'as good as' a fibration — i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration. (See homotopy fiber.)

The fibers are by definition the subspaces of E that are the inverse images of points b of B. If the base space B is path connected, it is a consequence of the definition that the fibers of two different points b₁ and b₂ in B are homotopy equivalent. Therefore one usually speaks of "the fiber" F.