Path (topology)

In mathematics, a path in a topological space X is a continuous map f from the unit interval I = to X

Related Topics:
Mathematics - Topological space - Continuous map - Unit interval

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: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.

Related Topics:
Curve - Parametrization

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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 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);

Related Topics:
Path-connected - Path-connected component

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:fg(s) = egin{cases}f(2s) & 0leq s leq rac{1}{2} \ g(2s-1) & rac{1}{2} leq s leq 1end{cases}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Note that path composition by itself is not associative due to the difference in parametrization. It is associative up to homotopy, see below.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~