Path space (algebraic topology)

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,*)} .

Source: Wikipedia — Path space (algebraic topology) (CC BY-SA 4.0)