# Path (topology)

In mathematics, a **path** in a topological space *X* is a continuous function *f* from the unit interval *I* = [0,1] to *X*

*f*:*I*→*X*.

The *initial point* of the path is *f*(0) and the *terminal point* is *f*(1). One often speaks of a "path from *x* to *y*" where *x* and *y* are the initial and terminal points of the path. Note that a path is not just a subset of *X* which "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* ∈ *X* is a path from *x* to *x*. A loop may be equally well regarded as a map *f* : *I* → *X* with *f*(0) = *f*(1) or as a continuous map from the unit circle *S*^{1} to *X*

*f*:*S*^{1}→*X*.

This is because *S*^{1} may be regarded as a quotient of *I* under the identification 0 ∼ 1. The set of all loops in *X* forms a space called the loop space of *X*.

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 a set of path-connected components. The set of path-connected components of a space *X* is often denoted π_{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}.

## Homotopy of paths

{{#invoke:main|main}}

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} : *I* → *X* indexed by *I* such that

*f*_{t}(0) =*x*_{0}and*f*_{t}(1) =*x*_{1}are fixed.- the map
*F*:*I*×*I*→*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 **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 an obvious 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*:

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 π_{1}(*X*,*x*_{0}).

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* ≥ 0. A path *f* of this kind has a length |*f*| defined as *a*. Path composition is then defined as before with the following modification:

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 (|*f*|+|*g*|+|*h*|)/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*. 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.

## References

- Ronald Brown, Topology and groupoids, Booksurge PLC, (2006).

- Peter May, A concise course in algebraic topology, University of Chicago Press, (1999).

- James Raymond Munkres, Topology 2ed, Prentice Hall, (2000).