Loop (topology)

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 $S 1$ into $X$ , because $S 1$ 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]

References

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

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

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Loop (topology)

See also

References