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**.^{
[1]} 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*.^{
[2]} The path space can then be viewed as the pullback of along .^{
[1]}

The natural map is a fibration called the
path space fibration.^{
[3]}

- ^
^{a}^{b}Martin Frankland, Math 527 - Homotopy Theory - Fiber sequences **^**Davis & Kirk 2001, Definition 6.14.**^**Davis & Kirk 2001, Theorem 6.15. 2.

- Davis, James F.; Kirk, Paul (2001).
*Lecture Notes in Algebraic Topology*(PDF). Graduate Studies in Mathematics. Vol. 35. Providence, RI: American Mathematical Society. pp. xvi+367. doi: 10.1090/gsm/035. ISBN 0-8218-2160-1. MR 1841974.