In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X
- f : I → X.
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 : I → X with f(0) = f(1) or as a continuous map from the unit circle S1 to X
- f : S1 → X.
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 a set of 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 in x0. Likewise, a loop in X is one that is based at x0.
Homotopy of 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 in X is a family of paths ft : I → X such that
- ft(0) = x0 and ft(1) = x1 are fixed.
- the map F : I × I → X given by F(s, t) = ft(s) is continuous.
The paths f0 and f1 connected by a homotopy are said to homotopic. One can likewise define a homotopy of loops keeping the base point fixed.
The property of being homotopic defines 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 an obvious 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:
Path composition, whenever defined, is not associative due to the difference in parametrization. It is associative at the level of homotopy however. 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).
Fundamental groupoid
There is a categorical picture of paths which is sometimes useful. Any topological space X gives raise 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 X.
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Last updated on Friday October 10, 2008 at 13:27:36 PDT (GMT -0700)
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