# Path (topology)

Short description: Continuous function whose domain is a closed unit interval
The points traced by a path from $\displaystyle{ A }$ to $\displaystyle{ B }$ in $\displaystyle{ \mathbb{R}^2. }$ However, different paths can trace the same set of points.

In mathematics, a path in a topological space $\displaystyle{ X }$ is a continuous function from the closed unit interval $\displaystyle{ [0, 1] }$ into $\displaystyle{ X. }$

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 $\displaystyle{ X }$ is often denoted $\displaystyle{ \pi_0(X). }$

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If $\displaystyle{ X }$ is a topological space with basepoint $\displaystyle{ x_0, }$ then a path in $\displaystyle{ X }$ is one whose initial point is $\displaystyle{ x_0 }$. Likewise, a loop in $\displaystyle{ X }$ is one that is based at $\displaystyle{ x_0 }$.

## Definition

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

Importantly, a path is not just a subset of $\displaystyle{ X }$ that "looks like" a curve, it also includes a parameterization. For example, the maps $\displaystyle{ f(x) = x }$ and $\displaystyle{ g(x) = x^2 }$ represent two different paths from 0 to 1 on the real line.

A loop in a space $\displaystyle{ X }$ based at $\displaystyle{ x \in X }$ is a path from $\displaystyle{ x }$ to $\displaystyle{ x. }$ A loop may be equally well regarded as a map $\displaystyle{ f : [0, 1] \to X }$ with $\displaystyle{ f(0) = f(1) }$ or as a continuous map from the unit circle $\displaystyle{ S^1 }$ to $\displaystyle{ X }$

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

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

## Homotopy of paths

Main page: Homotopy
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 $\displaystyle{ X }$ is a family of paths $\displaystyle{ f_t : [0, 1] \to X }$ indexed by $\displaystyle{ I = [0, 1] }$ such that

• $\displaystyle{ f_t(0) = x_0 }$ and $\displaystyle{ f_t(1) = x_1 }$ are fixed.
• the map $\displaystyle{ F : [0, 1] \times [0, 1] \to X }$ given by $\displaystyle{ F(s, t) = f_t(s) }$ is continuous.

The paths $\displaystyle{ f_0 }$ and $\displaystyle{ f_1 }$ 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 $\displaystyle{ f }$ under this relation is called the homotopy class of $\displaystyle{ f, }$ often denoted $\displaystyle{ [f]. }$

## Path composition

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

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

Clearly path composition is only defined when the terminal point of $\displaystyle{ f }$ coincides with the initial point of $\displaystyle{ g. }$ If one considers all loops based at a point $\displaystyle{ x_0, }$ 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, $\displaystyle{ [(fg)h] = [f(gh)]. }$ Path composition defines a group structure on the set of homotopy classes of loops based at a point $\displaystyle{ x_0 }$ in $\displaystyle{ X. }$ The resultant group is called the fundamental group of $\displaystyle{ X }$ based at $\displaystyle{ x_0, }$ usually denoted $\displaystyle{ \pi_1\left(X, x_0\right). }$

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

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

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

## Fundamental groupoid

There is a categorical picture of paths which is sometimes useful. Any topological space $\displaystyle{ X }$ gives rise to a category where the objects are the points of $\displaystyle{ X }$ 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 $\displaystyle{ X. }$ Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point $\displaystyle{ x_0 }$ in $\displaystyle{ X }$ is just the fundamental group based at $\displaystyle{ x_0 }$. More generally, one can define the fundamental groupoid on any subset $\displaystyle{ A }$ of $\displaystyle{ X, }$ using homotopy classes of paths joining points of $\displaystyle{ A. }$ This is convenient for Van Kampen's Theorem.