loop (topology)

{{Short description|Topological path whose initial point is equal to its terminal point}}

File:Fundamental group torus2.png.]]

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 whose initial point is equal to its terminal point.{{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}}.

A loop may also be seen as a continuous map {{mvar|f}} from the pointed unit circle {{math|S{{sup|1}}}} into {{mvar|X}}, because {{math|S{{sup|1}}}} may be regarded as a 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}}.