# Path (topology)

In mathematics, a path in a topological space $X$ is a continuous function from the closed unit interval $[0,1]$ into $X.$  The points traced by a path from $A$ to $B$ in $\mathbb {R} ^{2}.$ However, different paths can trace the same set of points.

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. Any space may be broken up into path-connected components. The set of path-connected components of a space $X$ is often denoted $\pi _{0}(X).$ One can also define paths and loops in pointed spaces, 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 is $x_{0}$ . Likewise, a loop in $X$ is one that is based at $x_{0}$ .

## Definition

A curve in a topological space $X$  is a continuous function $f:J\to X$  from a non-empty and non-degenerate interval $J\subseteq \mathbb {R} .$  A path in $X$  is a curve $f:[a,b]\to X$  whose domain $[a,b]$  is a compact non-degenerate interval (meaning $a  are real numbers), where $f(a)$  is called the initial point of the path and $f(b)$  is called its terminal point. A path from $x$  to $y$  is a path whose initial point is $x$  and whose terminal point is $y.$  Every non-degenerate compact interval $[a,b]$  is homeomorphic to $[0,1],$  which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function $f:[0,1]\to X$  from the closed unit interval $I:=[0,1]$  into $X.$  An arc or C0-arc in $X$  is a path in $X$  that is also a topological embedding.

Importantly, a path is not just a subset of $X$  that "looks like" a curve, it also includes a parameterization. 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 $x\in X$  is a path from $x$  to $x.$  A loop may be equally well regarded as a map $f:[0,1]\to X$  with $f(0)=f(1)$  or as a continuous map from the unit circle $S^{1}$  to $X$

$f:S^{1}\to X.$

This is because $S^{1}$  is the quotient space of $I=[0,1]$  when $0$  is identified with $1.$  The set of all loops in $X$  forms a space called the loop space of $X.$

## Homotopy of 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 $X$  is a family of paths $f_{t}:[0,1]\to X$  indexed by $I=[0,1]$  such that

• $f_{t}(0)=x_{0}$  and $f_{t}(1)=x_{1}$  are fixed.
• the map $F:[0,1]\times [0,1]\to X$  given by $F(s,t)=f_{t}(s)$  is continuous.

The paths $f_{0}$  and $f_{1}$  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 $f$  under this relation is called the homotopy class of $f,$  often denoted $[f].$

## Path composition

One can compose paths in a topological space in the following manner. Suppose $f$  is a path from $x$  to $y$  and $g$  is a path from $y$  to $z$ . The path $fg$  is defined as the path obtained by first traversing $f$  and then traversing $g$ :

$fg(s)={\begin{cases}f(2s)&0\leq s\leq {\frac {1}{2}}\\g(2s-1)&{\frac {1}{2}}\leq s\leq 1.\end{cases}}$

Clearly path composition is only defined when the terminal point of $f$  coincides with the initial point of $g.$  If one considers all loops based at a point $x_{0},$  then path composition is a binary operation.

Path composition, whenever defined, is not associative due to the difference in parametrization. However it is associative up to path-homotopy. That is, $[(fg)h]=[f(gh)].$  Path composition defines a group structure on the set of homotopy classes of loops based at a point $x_{0}$  in $X.$  The resultant group is called the fundamental group of $X$  based at $x_{0},$  usually denoted $\pi _{1}\left(X,x_{0}\right).$

In situations calling for associativity of path composition "on the nose," a path in $X$  may instead be defined as a continuous map from an interval $[0,a]$  to $X$  for any real $a\geq 0.$  (Such a path is called a Moore path.) A path $f$  of this kind has a length $|f|$  defined as $a.$  Path composition is then defined as before with the following modification:

$fg(s)={\begin{cases}f(s)&0\leq s\leq |f|\\g(s-|f|)&|f|\leq s\leq |f|+|g|\end{cases}}$

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

## Fundamental groupoid

There is a categorical picture of paths which is sometimes useful. Any topological space $X$  gives rise to a category where the objects are the points of $X$  and the morphisms 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 $X.$  Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point $x_{0}$  in $X$  is just the fundamental group based at $x_{0}$ . More generally, one can define the fundamental groupoid on any subset $A$  of $X,$  using homotopy classes of paths joining points of $A.$  This is convenient for the Van Kampen's Theorem.