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)