Path (topology)

In mathematics, a path in a topological space is a continuous function from the closed unit interval into

The points traced by a path from to in However, different paths can trace the same set of points.

Paths play an important role in the fields of topology and mathematical analysis. For example, 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 is often denoted

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If is a topological space with basepoint then a path in is one whose initial point is . Likewise, a loop in is one that is based at .


A curve in a topological space   is a continuous function   from a non-empty and non-degenerate interval   A path in   is a curve   whose domain   is a compact non-degenerate interval (meaning   are real numbers), where   is called the initial point of the path and   is called its terminal point. A path from   to   is a path whose initial point is   and whose terminal point is   Every non-degenerate compact interval   is homeomorphic to   which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function   from the closed unit interval   into   An arc or C0-arc in   is a path in   that is also a topological embedding.

Importantly, a path is not just a subset of   that "looks like" a curve, it also includes a parameterization. For example, the maps   and   represent two different paths from 0 to 1 on the real line.

A loop in a space   based at   is a path from   to   A loop may be equally well regarded as a map   with   or as a continuous map from the unit circle   to  


This is because   is the quotient space of   when   is identified with   The set of all loops in   forms a space called the loop space of  

Homotopy of pathsEdit

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   is a family of paths   indexed by   such that

  •   and   are fixed.
  • the map   given by   is continuous.

The paths   and   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   under this relation is called the homotopy class of   often denoted  

Path compositionEdit

One can compose paths in a topological space in the following manner. Suppose   is a path from   to   and   is a path from   to  . The path   is defined as the path obtained by first traversing   and then traversing  :


Clearly path composition is only defined when the terminal point of   coincides with the initial point of   If one considers all loops based at a point   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,   Path composition defines a group structure on the set of homotopy classes of loops based at a point   in   The resultant group is called the fundamental group of   based at   usually denoted  

In situations calling for associativity of path composition "on the nose," a path in   may instead be defined as a continuous map from an interval   to   for any real   (Such a path is called a Moore path.) A path   of this kind has a length   defined as   Path composition is then defined as before with the following modification:


Whereas with the previous definition,    , and   all have length   (the length of the domain of the map), this definition makes   What made associativity fail for the previous definition is that although   and  have the same length, namely   the midpoint of   occurred between   and   whereas the midpoint of   occurred between   and  . With this modified definition   and   have the same length, namely   and the same midpoint, found at   in both   and  ; more generally they have the same parametrization throughout.

Fundamental groupoidEdit

There is a categorical picture of paths which is sometimes useful. Any topological space   gives rise to a category where the objects are the points of   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   Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point   in   is just the fundamental group based at  . More generally, one can define the fundamental groupoid on any subset   of   using homotopy classes of paths joining points of   This is convenient for the Van Kampen's Theorem.

See alsoEdit


  • 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).