In mathematics, a path in a topological space is a continuous function from the closed unit interval into
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
Main article: Homotopy
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
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
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 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.
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.