In algebraic topology, a branch of mathematics, the based path space P X {\displaystyle PX} of a pointed space ( X , ∗ ) {\displaystyle (X,*)} is the space that consists of all maps f {\displaystyle f} from the interval I = [ 0 , 1 ] {\displaystyle I=[0,1]} to X such that f ( 0 ) = ∗ {\displaystyle f(0)=*} , called based paths. In other words, it is the mapping space from ( I , 0 ) {\displaystyle (I,0)} to ( X , ∗ ) {\displaystyle (X,*)} . A space X I {\displaystyle X^{I}} of all maps from I {\displaystyle I} to X, with no distinguished point for the start of the paths, is called the free path space of X. The maps from I {\displaystyle I} to X are called free paths. The path space P X {\displaystyle PX} is then the pullback of X I → X , χ ↦ χ ( 0 ) {\displaystyle X^{I}\to X,\,\chi \mapsto \chi (0)} along ∗ ↪ X {\displaystyle *\hookrightarrow X} . The natural map P X → X , χ → χ ( 1 ) {\displaystyle PX\to X,\,\chi \to \chi (1)} is a fibration called the path space fibration.
See also Mapping space
References
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.
Further reading https://ncatlab.org/nlab/show/path+space