Proof That Two Loops Can Be Joined to Make a New Loop
Theorem
Let
be the set of all loops in
with base point
For loops
define an operation
as follows:
Then
Proof
is formed by travelling around the loops
and then the loop
is continuous on
sinceis continuous on
and
is continuous on
and at
(
is the start and end point for each loop).
Also
and
Hence
