Path (topology)

In mathematics, a path in a topological space

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.

The set of path-connected components of a space

One can also define paths and loops in pointed spaces, which are important in homotopy theory.

is a compact non-degenerate interval (meaning

is called the initial point of the path and

which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function

represent two different paths from 0 to 1 on the real line.

A loop may be equally well regarded as a map

or as a continuous map from the unit circle

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.

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.

under this relation is called the homotopy class of

One can compose paths in a topological space in the following manner.

is defined as the path obtained by first traversing

: Clearly path composition is only defined when the terminal point of

then path composition is a binary operation.

Path composition, whenever defined, is not associative due to the difference in parametrization.

Path composition defines a group structure on the set of homotopy classes of loops based at a point

may instead be defined as a continuous map from an interval

Path composition is then defined as before with the following modification: Whereas with the previous definition,

(the length of the domain of the map), this definition makes

What made associativity fail for the previous definition is that although

gives rise to a category where the objects are the points of

and the morphisms are the homotopy classes of paths.

More generally, one can define the fundamental groupoid on any subset

using homotopy classes of paths joining points of