Path (topology)

Short description: Continuous function whose domain is a closed unit interval

This section appears to contradict another section of this article on the parameter interval on which a path is defined. Please see the talk page for more information. (June 2021)

|

The points traced by a path from A to B in ℝ2. However, different paths can trace the same set of points.

In mathematics, a path in a topological space X is a continuous function from the closed unit interval [0,1] into X.

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 X is often denoted π0(X).

One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x0, then a path in X is one whose initial point is x0. Likewise, a loop in X is one that is based at x0.

## Contents

* 1 Definition
* 2 Homotopy of paths
* 3 Path composition
* 4 Fundamental groupoid
* 5 See also
* 6 References

## Definition

A curve in a topological space X is a continuous function f:J->X from a non-empty and non-degenerate interval J⊆ℝ. A path in X is a curve f:[a,b]->X whose domain [a,b] is a compact non-degenerate interval (meaning a<b are real numbers), where f(a) is called the initial point of the path and f(b) is called its terminal point. A path from x to y is a path whose initial point is x and whose terminal point is y. Every non-degenerate compact interval [a,b] is homeomorphic to [0,1], which is why a path is sometimes, especially in homotopy theory, defined to be a continuous function f:[0,1]->X from the closed unit interval I:=[0,1] into X. An arc or C0-arc in X is a path in X that is also a topological embedding.

Importantly, a path is not just a subset of X that "looks like" a curve, it also includes a parameterization. For example, the maps f(x)=x and g(x)=x2 represent two different paths from 0 to 1 on the real line.

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:[0,1]->X with f(0)=f(1) or as a continuous map from the unit circle S1 to X

f:S1->X.

This is because S1 is the quotient space of I=[0,1] when 0 is identified with 1. The set of all loops in X forms a space called the loop space of X.

## Homotopy of paths

Main article: Homotopy

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 X is a family of paths ft:[0,1]->X indexed by I=[0,1] such that

* ft(0)=x0 and ft(1)=x1 are fixed.
* the map F:[0,1]×[0,1]->X given by F(s,t)=ft(s) is continuous.

The paths f0 and f1 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 f under this relation is called the homotopy class of f, often denoted [f].

## Path composition

One can compose paths in a topological space in the following 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:

fg(s)={f(2s)0≤s≤12g(2s−1)12≤s≤1.

Clearly path composition is only defined when the terminal point of f coincides with the initial point of g. If one considers all loops based at a point x0, 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, [(fg)h]=[f(gh)]. Path composition defines a group structure on the set of homotopy classes of loops based at a point x0 in X. The resultant group is called the fundamental group of X based at x0, usually denoted π1(X,x0).

In situations calling for associativity of path composition "on the nose," a path in X may instead be defined as a continuous map from an interval [0,a] to X for any real a≥0. (Such a path is called a Moore path.) A path f of this kind has a length |f| defined as a. Path composition is then defined as before with the following modification:

fg(s)={f(s)0≤s≤|f|g(s−|f|)|f|≤s≤|f|+|g|

Whereas with the previous definition, f, g, and fg all have length 1 (the length of the domain of the map), this definition makes |fg|=|f|+|g|. What made associativity fail for the previous definition is that although (fg)h and f(gh)have the same length, namely 1, the midpoint of (fg)h occurred between g and h, whereas the midpoint of f(gh) occurred between f and g. With this modified definition (fg)h and f(gh) have the same length, namely |f|+|g|+|h|, and the same midpoint, found at (|f|+|g|+|h|)/2 in both (fg)h and f(gh); more generally they have the same parametrization throughout.

## Fundamental groupoid

There is a categorical picture of paths which is sometimes useful. Any topological space X gives rise to a category where the objects are the points of X 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 X. Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point x0 in X is just the fundamental group based at x0. More generally, one can define the fundamental groupoid on any subset A of X, using homotopy classes of paths joining points of A. This is convenient for Van Kampen's Theorem.

## See also

* Curve § Topology
* Path space (disambiguation)

## References

* 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).

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Path (topology). Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Path_(topology)&oldid=189519"