Given a topological space
and a point
let
be the set of loops in
with base point
is an equivalence relation on a set
for the set of loops with the same base point
with
if
is a loop inhomotopic to a loop
andand
have the same base point
The set of homotopy classes is then
The operation
of composing loops is associative and has identity
Also ifis a loop,
and since
the inverse of
is
The space
with operationis a group with identity
called the fundamental group of the spacebased on