Proof That The Homotopy Property With Given Base Point is an Equivalence Relation
Theorem
Let
be the set of all loops in
with base point
The relation 'homotopic relative to
labelled by
defined onis an equivalence relation.
Proof
since we can define
such that
for all
since if
there exists a continuous functio
such that
so define
then
and 
so
is continuous since
is.
Ifthere exists
such that
if
there exists
such that
Define
Then
and
and
is continuous hence