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.
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.
See also
- Free loop
- Loop group
- Loop space
- Loop algebra
- Fundamental group
- Quasigroup
References
- Adams, John Frank (1978), Infinite Loop Spaces, Annals of mathematics studies, vol. 90, Princeton University Press, p. 3, ISBN 9780691082066.
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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 Two loops a b in a torus 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 See alsoFree loop Loop group Loop space Loop algebra Fundamental group QuasigroupReferencesAdams John Frank 1978 Infinite Loop Spaces Annals of mathematics studies vol 90 Princeton University Press p 3 ISBN 9780691082066 This topology related article is a stub You can help Wikipedia by expanding it vte
