Path (topology)
Get Path Topology essential facts below. View Videos or join the Path Topology discussion. Add Path Topology to your PopFlock.com topic list for future reference or share this resource on social media.
Path Topology
The points traced by a path from A to B in R². However, different paths can trace the same set of points.

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

f : IX.

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) = x2 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 S1 to X

f : S1X.

This is because S1 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 path-connected components. The set of path-connected components of a space X is often denoted ?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 x0, then a path in X is one whose initial point is x0. Likewise, a loop in X is one that is based at x0.

## Homotopy of paths

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 X is a family of paths ft : I -> X indexed by I such that

• ft(0) = x0 and ft(1) = x1 are fixed.
• the map F : I × I -> X given by F(s, t) = ft(s) is continuous.

The paths f0 and f1 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:

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

Clearly path composition is only defined when the terminal point of f coincides with the initial point of g. If one considers all loops based at a point x0, 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 x0 in X. The resultant group is called the fundamental group of X based at x0, usually denoted ?1(X,x0).

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:

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

Whereas with the previous definition, 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 x0 in X is just the fundamental group based at x0. 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).
• J. Peter May, A concise course in algebraic topology, University of Chicago Press, (1999).
• James Munkres, Topology 2ed, Prentice Hall, (2000).