A loop is a continuous mapping from the circle into
the order parameter space:
->
**M**(
),
0 <=
<=
2
When we encircle the defect with a loop, we get a
loop in order parameter space as shown in figure 4 (reproduced below):
->
**x**(
)
is the loop in real space, and
->
**M**(**x**(
))
is the loop in order parameter space. Two loops
are equivalent if there is a continuous one-parameter family of loops
connecting one to the other:
**M****N** if there exists
*M(t)*(
)
continuous both in
and in 0 <= *t* <= 1, with
*M(0)*(
)
**M**
and
*M(1)*(
)
**N**.

Figure 4 reproduced.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).