In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

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. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

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 Illustration of a unit circle. ...

f : S¹ → X.

This is because S¹ may be regarded as a quotient of I under the identification 0 ∼ 1. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...

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); In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

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:

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. In mathematics, associativity is a property that a binary operation can have. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, the space of loops or loop space of a topological space X is the topological space of continuous maps from the circle S1 to X with the compact-open topology. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed 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. Image File history File links made by me in Inkscape. ... Image File history File links made by me in Inkscape. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

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]. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

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). In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...