Path (topology)

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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 map 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 parametrization. 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 : S¹ → X.

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 a set of path-connected components. The set of path-connected components of a space X is often denoted π₀(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 in x₀. Likewise, a loop in X is one that is based at x₀.

[edit] Homotopy of paths

Main article: 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 in X is a family of paths f_t : I → X such that

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

The paths f₀ and f₁ connected by a homotopy are said to homotopic. One can likewise define a homotopy of loops keeping the base point fixed.

The property of being homotopic defines 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].

[edit] 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 x₀, then path composition is a binary operation.

Path composition, whenever defined, is not associative due to the difference in parametrization. It is associative at the level of homotopy however. 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 π₁(X,x₀).

[edit] Fundamental groupoid

There is a categorical picture of paths which is sometimes useful. Any topological space X gives raise 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 X.