Editing Loop (topology)

{{Short description|Topological path whose initial point is equal to its terminal point}}
[[File:Fundamental group torus2.png|thumb|Two loops {{mvar|a}}, {{mvar|b}} in a [[torus]].]]

In [[mathematics]], a '''loop''' in a [[topological space]] {{mvar|X}} is a [[continuous function]] {{mvar|f}} from the [[unit interval]] {{math|1=''I'' = [0,1]}} to {{mvar|X}} such that {{nowrap|{{math|1=''f''(0) = ''f''(1)}}.}} In other words, it is a [[path (topology)|path]] whose initial point is equal to its terminal point.<ref name="ils">{{citation|title=Infinite Loop Spaces|volume=90|series=Annals of mathematics studies|first=John Frank|last=Adams|authorlink = John Frank Adams|publisher=[[Princeton University Press]]|year=1978|isbn=9780691082066|page=3|url=https://books.google.com/books?id=e2rYkg9lGnsC&pg=PA3}}.</ref>

A loop may also be seen as a continuous map {{mvar|f}} from the [[Pointed space|pointed]] [[unit circle]] {{math|''S''{{sup|1}}}} into {{mvar|X}}, because {{math|''S''{{sup|1}}}} may be regarded as a [[Quotient space (topology)|quotient]] of {{mvar|I}} under the identification of 0 with 1.

The set of all loops in {{mvar|X}} forms a space called the [[loop space]] of {{mvar|X}}.<ref name="ils"/>

==See also==
*[[Free loop]]
*[[Loop group]]
*[[Loop space]]
*[[Loop algebra]]
*[[Fundamental group]]
*[[Quasigroup]]

==References==
{{reflist}}

{{DEFAULTSORT:Loop (Topology)}}
[[Category:Topology]]

[[es:Grupo fundamental#Lazo]]


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