Path
A continuous mapping 
of the interval 
into a topological space 
. The points 
and 
are called the initial and the final points of the path 
. Given 
, the path defined by the formula 
, 
, is called the path inverse to 
and is denoted by 
. Given 
and 
with 
, the path defined by the formula
 |
is called the composite of the paths 
and 
and is denoted by 
. In a path-connected space 
with distinguished point 
, the set of all paths with initial point 
forms the path space of 
.
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 
, the composition defined above becomes associative, and 
becomes a genuine inverse to 
. See Fundamental groupoid.
More precisely, one may define a path as being any continuous mapping 
, where 
is called the length of the path 
. Then 
and 
, with 
of length 
and 
, are composed to 
, taking 
to 
and 
in 
(where 
has length 
) to 
. 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) |
Path. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Path&oldid=48142