Proof that Equivalence Classes Are Well Defined For The Set of Loops On a Space
is an equivalence relation on a setfor the set of loops with the same base pointwithifis a loop inhomotopic to a loopandandhave the same base point
The set of homotopy classes is then
The homotopy class of the loopis written
With this definition,is well defined.
is a loop with base pointproved here, hence an element of
Ifthen a continuous functionexists such thatand