Editing Path (topology) (section)

{{short description|Continuous function whose domain is a closed unit interval}}
{{No footnotes|date=June 2020}}
[[Image:Path.svg|thumb|The points traced by a path from <math>A</math> to <math>B</math> in <math>\mathbb{R}^2.</math> However, different paths can trace the same set of points.]]
In [[mathematics]], a '''path''' in a [[topological space]] <math>X</math> is a [[Continuous function (topology)|continuous function]] from a [[closed interval]] into <math>X.</math> 

Paths play an important role in the fields of [[topology]] and [[mathematical analysis]]. 
For example, a topological space for which there exists a path connecting any two points is said to be [[Path-connected space|path-connected]]. Any space may be broken up into [[path-connected component]]s. The set of path-connected components of a space <math>X</math> is often denoted <math>\pi_0(X).</math> 

One can also define paths and loops in [[pointed space]]s, which are important in [[homotopy theory]]. If <math>X</math> is a topological space with basepoint <math>x_0,</math> then a path in <math>X</math> is one whose initial point is <math>x_0</math>. Likewise, a loop in <math>X</math> is one that is based at <math>x_0</math>.