Proof That The Homotopy Property With Given Base Point is an Equivalence Relation
Letbe the set of all loops inwith base pointThe relation 'homotopic relative tolabelled bydefined onis an equivalence relation.
since we can definesuch thatfor all
since ifthere exists a continuous functiosuch thatso definethenand sois continuous sinceis.
Ifthere existssuch thatifthere existssuch that
Thenandandis continuous hence