path topology

The points traced by a path from ''A'' to ''B'' in '''R'''². However, different paths can trace the same set of points.

In [mathematics] , a path in a [topological space] X is a [continuous map] f from the [unit interval] I = [0,1] to X
: f : I → X .
The initial point of the path is f (0) and the terminal point is f (1). One often speaks of a "path from x to y " where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a [curve] , it also includes a [parametrization] . For example, the maps f ( x ) = x and g ( x ) = x 2 represent two different paths from 0 to 1 on the real line.

A [loop] in a space X based at x ∈ X is a path from x to x . A loop may be equally well regarded as a map f : I → X with f (0) = f (1) or as a continuous map from the [unit circle] S 1 to X
: f : S 1 → X .
This is because S 1 may be regarded as a [quotient] of I under the identification 0 ∼ 1. The set of all loops in X forms a space called the [loop space] of X .

A topological space for which there exists a path connecting any two points is said to be [path-connected] . Any space may be broken up into a set of [path-connected component] s. The set of path-connected components of a space X is often denoted π0( X );.

One can also define paths and loops in [pointed space] s, which are important in [homotopy theory] . If X is a topological space with basepoint x 0, then a path in X is one whose initial point in x 0. Likewise, a loop in X is one that is based at x 0.

Homotopy of paths