Dyadic Solenoid
As an additive abelian group the dyadic rationals are the direct limit of infinite cyclic subgroups of the rational numbers,
In the spirit of Pontryagin duality, there is a dual object, namely the inverse limit of the unit circle group under the repeated squaring map
The resulting dual is a topological group D called the dyadic solenoid, an example of a solenoid group.
An element of the dyadic solenoid can be represented as an infinite sequence of complex numbers q0, q1, q2, ..., with the properties that each qi lies on the unit circle and that, for all i > 0, qi2 = qi-1. The group operation on these elements multiplies any two sequences componentwise.
As a topological space the dyadic solenoid is an indecomposable continuum.
Read more about this topic: Dyadic Rational