Loop (topology)

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Two loops a, b in a torus.

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.[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]

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