Fibration

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.