View source for Path (topology)

{{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>.

== 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>

== Homotopy of paths ==
{{Main|Homotopy}}
[[Image:Homotopy between two paths.svg|thumb|right|A homotopy between two paths.]]

Paths and loops are central subjects of study in the branch of [[algebraic topology]] called [[homotopy theory]]. A [[homotopy]] of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.

Specifically, a homotopy of paths, or '''path-homotopy''', in <math>X</math> is a family of paths <math>f_t : [0, 1] \to X</math> indexed by <math>I = [0, 1]</math> such that
* <math>f_t(0) = x_0</math> and <math>f_t(1) = x_1</math> are fixed. 
* the map <math>F : [0, 1] \times [0, 1] \to X</math> given by <math>F(s, t) = f_t(s)</math> is continuous. 
The paths <math>f_0</math> and <math>f_1</math> connected by a homotopy are said to be '''homotopic''' (or more precisely '''path-homotopic''', to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.

The relation of being homotopic is an [[equivalence relation]] on paths in a topological space. The [[equivalence class]] of a path <math>f</math> under this relation is called the '''homotopy class''' of <math>f,</math> often denoted <math>[f].</math> 

== Path composition ==

One can compose paths in a topological space in the following manner. Suppose <math>f</math> is a path from <math>x</math> to <math>y</math> and <math>g</math> is a path from <math>y</math> to <math>z</math>. The path <math>fg</math> is defined as the path obtained by first traversing <math>f</math> and then traversing <math>g</math>:
:<math>fg(s) = \begin{cases}f(2s) & 0 \leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases}</math>
Clearly path composition is only defined when the terminal point of <math>f</math> coincides with the initial point of <math>g.</math> If one considers all loops based at a point <math>x_0,</math> then path composition is a [[binary operation]]. 

Path composition, whenever defined, is not [[associative]] due to the difference in parametrization. However it {{em|is}} associative up to path-homotopy. That is, <math>[(fg)h] = [f(gh)].</math> Path composition defines a [[Group (mathematics)|group structure]] on the set of homotopy classes of loops based at a point <math>x_0</math> in <math>X.</math> The resultant group is called the [[fundamental group]] of <math>X</math> based at <math>x_0,</math> usually denoted <math>\pi_1\left(X, x_0\right).</math> 

In situations calling for associativity of path composition "on the nose," a path in <math>X</math> may instead be defined as a continuous map from an interval <math>[0, a]</math> to <math>X</math> for any real <math>a \geq 0.</math> (Such a path is called a [[Moore path]].) A path <math>f</math> of this kind has a length <math>|f|</math> defined as <math>a.</math>  Path composition is then defined as before with the following modification:
:<math>fg(s) = \begin{cases}f(s) & 0 \leq s \leq |f| \\ g(s-|f|) & |f| \leq s \leq |f| + |g|\end{cases}</math>

Whereas with the previous definition, <math>f,</math> <math>g</math>, and <math>fg</math> all have length <math>1</math> (the length of the domain of the map), this definition makes <math>|fg| = |f| + |g|.</math> What made associativity fail for the previous definition is that although <math>(fg)h</math> and <math>f(gh)</math>have the same length, namely <math>1,</math> the midpoint of <math>(fg)h</math> occurred between <math>g</math> and <math>h,</math> whereas the midpoint of <math>f(gh)</math> occurred between <math>f</math> and <math>g</math>.  With this modified definition <math>(fg)h</math> and <math>f(gh)</math> have the same length, namely <math>|f| + |g| + |h|,</math> and the same midpoint, found at <math>\left(|f| + |g| + |h|\right)/2</math> in both <math>(fg)h</math> and <math>f(gh)</math>; more generally they have the same parametrization throughout.

== Fundamental groupoid ==

There is a [[Category theory|categorical]] picture of paths which is sometimes useful. Any topological space <math>X</math> gives rise to a [[Category (mathematics)|category]] where the objects are the points of <math>X</math> and the [[morphism]]s are the homotopy classes of paths. Since any morphism in this category is an [[isomorphism]], this category is a [[groupoid]] called the [[fundamental groupoid]] of <math>X.</math> Loops in this category are the [[endomorphism]]s (all of which are actually [[automorphism]]s). The [[automorphism group]] of a point <math>x_0</math> in <math>X</math> is just the fundamental group based at <math>x_0</math>. More generally, one can define the fundamental groupoid on any subset <math>A</math> of <math>X,</math> using homotopy classes of paths joining points of <math>A.</math> This is convenient for [[Van Kampen's Theorem]].

== See also ==
* {{Section link|Curve#Topology}}
* {{annotated link|Locally path-connected space}}
* [[Path space (disambiguation)]]<!--intentional link to DAB page-->
* {{annotated link|Path-connected space}}

== References ==

* [[Ronald Brown (mathematician)|Ronald Brown]], Topology and groupoids, Booksurge PLC, (2006).
* [[J. Peter May]], A concise course in algebraic topology, University of Chicago Press, (1999).
* [[James Munkres]], Topology  2ed, Prentice Hall, (2000).

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[[Category:Topology]]
[[Category:Homotopy theory]]