Solenoid

Example

Let $p$ be a prime and let $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ be the circle group. The $p$-adic solenoid $\Sigma_p$ is the inverse limit of the sequence $$\cdots \to \mathbb{T} \xrightarrow{z \mapsto z^p} \mathbb{T} \xrightarrow{z \mapsto z^p} \mathbb{T}.$$ Its elements are sequences $(z_0, z_1, z_2, \dots)$ in $\mathbb{T}$ with $z_{k+1}^p = z_k$ for all $k$. The group law is coordinatewise multiplication, and the topology is the one induced from the product $\mathbb{T}^{\N}$.

Check that it is a compact group. Each map $z \mapsto z^p$ is a continuous homomorphism, so the sequences with $z_{k+1}^p = z_k$ form a subgroup of $\mathbb{T}^{\N}$. This subgroup is closed, since it is cut out by the equations $z_{k+1}^p = z_k$ between continuous functions. The product $\mathbb{T}^{\N}$ is compact by Tychonoff's theorem, so $\Sigma_p$ is a compact Hausdorff abelian group.