Path (topology)

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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.

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 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:

[fg(s) = \beginf(2s) & 0\leq s \leq \frac \\ g(2s-1) & \frac \leq s \leq 1\end]

Note that path composition by itself is not associative due to the difference in parametrization. The algebraic object formed by the set of all loops in a space, together with the operation of path composition, is a quasigroup called the loop space. A group can be formed by considering homotopy equivalence classes, as below.

Homotopy theory

A homotopy between two paths.

Paths and loops are extremely important in 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].

Although path composition is not associative at the level of paths, it is associative at the level of 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 in X. The resultant group is called the fundamental group of X based at x, usually denoted π₁(X,x).

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