Loop (topology)

Two loops a, b in a torus.

A loop in mathematics, 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 the terminal point.^[1]

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.^[1]

See also

• Free loop
• Loop group
• Loop space
• Loop algebra
• Fundamental group
• Quasigroup

References

1. Adams, John Frank (1978), Infinite Loop Spaces, Annals of mathematics studies, 90, Princeton University Press, p. 3, ISBN 9780691082066.