loop
A loop based at in a topological space is simply a continuous map with .
The collection of all such loops, modulo homotopy equivalence, forms a group known as the fundamental group
.
More generally, the space of loops in based at with the compact-open topology, represented by , is known as the loop space
of . And one has the homotopy groups
, where represents the higher homotopy groups, and is the basepoint in consisting of the constant loop at .