construction, any continuous mapping can be extended to a fibration by enlarging its domain to a homotopy equivalent space. This fibration is called pathspace fibration. The total space of the pathspace fibration for a continuous mapping between topological spaces consists of pairs with and paths with starting point where is the unit interval. The space carries the subspace topology of where describes the space of all mappings and carries the compact-open topology. The pathspace fibration is given by the mapping with The fiber is also called the homotopy fiber of and consists of the pairs with and paths where and holds. For the special case of the inclusion of the base point , an important example of the pathspace fibration emerges. The total space consists of all paths in which starts at This space is denoted by and is called path space. The pathspace fibration maps each path to its endpoint, hence the fiber consists of all closed paths. The fiber is denoted by and is called loop space. Properties The fibers over are homotopy equivalent for each path component of For a homotopy the pullback fibrations and are fiber homotopy equivalent. If the base space is contractible, then the fibration is fiber homotopy equivalent to the product fibration The pathspace fibration of a fibration is very similar to itself. More precisely, the inclusion is a fiber homotopy equivalence. For a fibration with fiber and contractible total space, there is a weak homotopy equivalence Puppe sequence For a fibration with fiber and base point the inclusion of the fiber into the homotopy fiber is a homotopy equivalence. The mapping with , where and is a path from to in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration . This procedure can now be applied again to the fibration and so on. This leads to a long sequence:The fiber of over a point consists of the pairs with closed paths and starting point , i.e. the loop space . The inclusion is a homotopy equivalence and iteration yields the sequence:Due to the duality of fibration and cofibration, there also exists a sequence of cofibrations. These two sequences are known as the Puppe sequences or the sequences of fibrations and cofibrations. Principal fibration A fibration with fiber is called principal, if there exists a commutative diagram: The bottom row is a sequence of fibrations