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.

REFERENCES:

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 323, 1976.