Loop (topology)

Short description: Topological path whose initial point is equal to its terminal point

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 $S 1$ 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

↑ ^1.0 ^1.1 Adams, John Frank (1978), Infinite Loop Spaces, Annals of mathematics studies, 90, Princeton University Press, p. 3, ISBN 9780691082066, https://books.google.com/books?id=e2rYkg9lGnsC&pg=PA3 .

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[ils-1] 1.0 ^1.1 Adams, John Frank (1978), Infinite Loop Spaces, Annals of mathematics studies, 90, Princeton University Press, p. 3, ISBN 9780691082066, https://books.google.com/books?id=e2rYkg9lGnsC&pg=PA3 .

[1]

Loop (topology)

See also

References