P Adic Solenoid

The canonical group homomorphisms

$\varphi_n:\mathbb{R}/p^{n+1}\mathbb{Z}\to \mathbb{R}/p^n\mathbb{Z},~x\pmod{p^{n+1}\mathbb{Z}}\mapsto x\pmod{p^n\mathbb{Z}}$

form a projective system $(\mathbb{R}/p^n\mathbb{Z},\varphi_n)_{n\ge 0}$ of topological groups. The $p$ adic solenoid $\mathbb{S}_p$ is a projective limit $\mathbb{S}_p=\lim_{\longleftarrow}\mathbb{R}/p^n\mathbb{Z}$ of the projective system $(\mathbb{R}/p^n\mathbb{Z},\varphi_n)_{n\ge 0}$ . It is both compact and connected.

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P Adic Solenoid