In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). 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 f from the pointed unit circle S1 into X, because S1 may be regarded as a quotient of I under the identification of 0 with 1. The set of all loops in X forms a space called the loop space of X.
Definition Let X {\displaystyle X} be a topological space. A loop is a continuous function f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X} such that f ( 0 ) = f ( 1 ) {\displaystyle f(0)=f(1)} . If f {\displaystyle f} begins and ends at x 0 ∈ X {\displaystyle x_{0}\in X} the loop is said to be based at x 0 {\displaystyle x_{0}} . A loop is then a path that begins and ends at the same point x 0 {\displaystyle x_{0}} . The set of homotopy classes of loops based at x 0 {\displaystyle x_{0}} together with the operation of path composition, forms the fundamental group of X {\displaystyle X} relative to x 0 {\displaystyle x_{0}} , usually denoted by π 1 ( X , x 0 ) {\displaystyle \pi _{1}(X,x_{0})} .
See also Free loop Loop group Loop space Loop algebra Fundamental group Quasigroup
References