Loop (topology)



In mathematics, a loop in a topological space $a$ is a continuous function $b$ 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 $f$ under the identification of 0 with 1.

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