## The Fundamental Group of the Space (X,T)

Given a topological spaceand a pointletbe the set of loops in with base point

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 operationof composing loops is associative and has identity

Also ifis a loop,and sincethe inverse ofis

The spacewith operationis a group with identitycalled the fundamental group of the spacebased on