Editing Path (topology) (section)

== Definition ==

A ''[[Topological curve|curve]]'' in a [[topological space]] <math>X</math> is a [[Continuous function (topology)|continuous function]] <math>f : J \to X</math> from a non-empty and [[non-degenerate interval]] <math>J \subseteq \R.</math> 
A '''{{em|path}}''' in <math>X</math> is a curve <math>f : [a, b] \to X</math> whose domain <math>[a, b]</math> is a [[Compact space|compact]] non-degenerate interval (meaning <math>a < b</math> are [[real number]]s), where <math>f(a)</math> is called the '''{{em|initial point}}''' of the path and <math>f(b)</math> is called its '''{{em|terminal point}}'''. 
A '''{{em|path from <math>x</math> to <math>y</math>}}''' is a path whose initial point is <math>x</math> and whose terminal point is <math>y.</math> 
Every non-degenerate compact interval <math>[a, b]</math> is [[homeomorphic]] to <math>[0, 1],</math> which is why a '''{{em|path}}''' is sometimes, especially in homotopy theory, defined to be a [[Continuous function (topology)|continuous function]] <math>f : [0, 1] \to X</math> from the closed [[unit interval]] <math>I := [0, 1]</math> into <math>X.</math>

{{anchor|Arc|C0 arc}}
An '''{{em|arc}}''' or '''{{mvar|C}}<sup>0</sup>{{em|-arc}}''' in <math>X</math> is a path in <math>X</math> that is also a [[topological embedding]]. 

Importantly, a path is not just a subset of <math>X</math> that "looks like" a [[Topological curve|curve]], it also includes a [[Parametrization (geometry)|parameterization]]. For example, the maps <math>f(x) = x</math> and <math>g(x) = x^2</math> represent two different paths from 0 to 1 on the real line. 

A '''[[Loop (topology)|loop]]''' in a space <math>X</math> based at <math>x \in X</math> is a path from <math>x</math> to <math>x.</math> A loop may be equally well regarded as a map <math>f : [0, 1] \to X</math> with <math>f(0) = f(1)</math> or as a continuous map from the [[unit circle]] <math>S^1</math> to <math>X</math>
:<math>f : S^1 \to X.</math>
This is because <math>S^1</math> is the [[Quotient space (topology)|quotient space]] of <math>I = [0, 1]</math> when <math>0</math> is identified with <math>1.</math> The set of all loops in <math>X</math> forms a space called the [[loop space]] of <math>X.</math>