Loop (topology)

File:Fundamental group torus2.png

Two loops Template:Mvar, Template:Mvar in a torus.

In mathematics, a loop in a topological space Template:Mvar is a continuous function Template:Mvar from the unit interval I = [0,1] to Template:Mvar 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 Template:Mvar from the pointed unit circle S¹ into Template:Mvar, because S¹ may be regarded as a quotient of Template:Mvar under the identification of 0 with 1.

The set of all loops in Template:Mvar forms a space called the loop space of Template:Mvar.

See also

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

References

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