# Loop (topology)

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 *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]}

## See also

## References

- 1 2 Adams, John Frank (1978),
*Infinite Loop Spaces*, Annals of mathematics studies,**90**, Princeton University Press, p. 3, ISBN 9780691082066.

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