In algebraic topology, the path space fibration over a based space is a fibration of the form where * , equipped with the compact-open topology, is the space called the of X, * is the fiber of over the base point of X; thus it is the loop space of X. The space consists of all maps from I to X that may not preserve the base points; it is called the free path space of X and the fibration given by, say, , is called the free path space fibration.