In algebraic topology, a branch of mathematics, the based path space of a pointed space is the space that consists of all maps from the interval to X such that , called based paths. In other words, it is the mapping space from to .
A space of all maps from to X, with no distinguished point for the start of the paths, is called the free path space of X. The maps from to X are called free paths. The path space is then the pullback of along .
The natural map is a fibration called the path space fibration.