If is a Fiber Bundle with a Paracompact Topological Space, then satisfies the Homotopy Lifting Property with respect to all Topological Spaces. In other words, if is a Homotopy from to , and if is a Lift of the Map with respect to , then has a Lift to a Map with respect to . Therefore, if you have a Homotopy of a Map into , and if the beginning of it has a Lift, then that Lift can be extended to a Lift of the Homotopy itself.
A fibration is a Map between Topological Spaces such that it satisfies the Homotopy Lifting Property.
See also Fiber Bundle, Fiber Space