Path space (algebraic topology)
In algebraic topology, a branch of mathematics, the path space  of a based space  is the space that consists of all maps  from the interval  to X such that , called paths. In other words, it is the mapping space from  to . The space  of all maps from  to X (free paths or just paths) is called the free path space of X. The path space  can then be viewed as the pullback of  along . The natural map  is a fibration called the path space fibration. References

Further reading 
https://ncatlab.org/nlab/show/path+space

Algebraic topology