Loop (topology)

thumb|Two loops , in a [[torus.]]

In mathematics, a loop in a topological space is a continuous function from the unit interval to such that In other words, it is a path whose initial point is equal to its terminal point.

A loop may also be seen as a continuous map from the pointed unit circle into , because may be regarded as a quotient of under the identification of 0 with 1.

The set of all loops in forms a space called the loop space of .

The set of homotopy classes of loops based at <math>x_0</math> together with the operation of path composition, forms the fundamental group of <math>X</math> relative to <math>x_0</math>, usually denoted by <math>\pi_1(X,x_0)</math>.