Path

A continuous mapping $f$ of the interval $[ 0 , 1 ]$ into a topological space $X$. The points $f ( 0)$ and $f ( 1)$ are called the initial and the final points of the path $f$. Given $f$, the path defined by the formula $t \rightarrow f ( 1- t )$, $t \in [ 0 , 1 ]$, is called the path inverse to $f$ and is denoted by $f ^ { - 1 }$. Given $f _ {1}$ and $f _ {2}$ with $f _ {1} ( 1) = f _ {2} ( 0)$, the path defined by the formula

$$t \rightarrow \left \{ \begin{array}{ll} {f _ {1} ( 2t ) , } &{ t _ {2} \leq 1/2 , } \\ {f( 2t- 1), } &{ t \geq 1/2 , } \\ \end{array} \right .$$

is called the composite of the paths $f _ {1}$ and $f _ {2}$ and is denoted by $f _ {1} f _ {2}$. In a path-connected space $X$ with distinguished point $*$, the set of all paths with initial point $*$ forms the path space of $X$.

Comments

Generally one is interested not so much in the individual paths in a space as in the homotopy classes thereof; if one factors by the equivalence relation of homotopy relative to $\{ 0, 1 \}$, the composition defined above becomes associative, and $f ^ { - 1 }$ becomes a genuine inverse to $f$. See Fundamental groupoid.

More precisely, one may define a path as being any continuous mapping $f: [ 0, r] \rightarrow X$, where $r \geq 0$ is called the length of the path $f$. Then $f _ {1}$ and $f _ {2}$, with $f _ {1}$ of length $r$ and $f _ {2} ( 0) = f _ {1} ( r)$, are composed to $f _ {1} f _ {2}$, taking $t \leq r$ to $f _ {1} ( t)$ and $t$ in $[ r, r+ s]$( where $f _ {2}$ has length $s$) to $f _ {2} ( t- r)$. This composition is associative (not only homotopy associative).

References

[a1]	P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1965)