Path (topology)
[Image:Path.svg|thumb|The points traced by a path from to in However, different paths can trace the same set of points.]
In mathematics, a path in a topological space is a continuous function from a closed 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.
Definition
A curve in a topological space is a continuous function from a non-empty and non-degenerate intervalA ' in is a curve whose domain is a compact non-degenerate interval, where is called the ' of the path and is called its '.
A ' is a path whose initial point is and whose terminal point is
Every non-degenerate compact interval is homeomorphic to which is why a ' is sometimes, especially in homotopy theory, defined to be a continuous function from the closed unit interval into
An ' or 0 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 paths
[Image:Homotopy between two paths.svg|thumb|right|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 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 composition
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 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, 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.