Loop (topology)

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.

Definition

Let be a topological space. A loop is a continuous function such that. If begins and ends at the loop is said to be based at . A loop is then a path that begins and ends at the same point.
The set of homotopy classes of loops based at together with the operation of path composition, forms the fundamental group of relative to , usually denoted by.