Theorem

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

Proof

Letbe any three equivalence classes in

By definition,

and

The homotopy betweenandis defined as