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Path (topology)

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 **

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