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- f : I → X.
The initial point of the path is f(0) and the terminal point is f(1). One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a curve, it also includes a parametrization. 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 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.
This is because S1 may be regarded as a quotient of I under the identification 0 ∼ 1.
A topological space for which the 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 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:
It is important to note that path composition is not associative due to problems with parametrization.
Paths and loops are extremely important in 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].
Although path composition is not associative at the level of paths, it is associative at the level of 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 x in X. The resultant group is called the fundamental group of X based at x, usually denoted π1(X,x).
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