Proof That Associativity is Preserved on Homotopy Classes


is an equivalence relation on a setfor the set of loops with the same base pointwithifisloop inhomotopic to a loopandandhave the same base pointThe set of homotopy classes is then

Associativity on the set of homotopy classes, writtenis preserved, so that


Letbe any three equivalence classes in

By definition,


The homotopy betweenandis defined as