Exercise 6, § 7

Let $(G_n, f_{nm})_{n \in \mathbf{N}}$ be the projective system of compact groups such that $G_n = \mathbf{T} = \mathbf{R}/\mathbf{Z}$ for every $n$, and $f_{nm}$ is the continuous homomorphism $x \mapsto p^{m-n}x$ of $\mathbf{T}$ onto itself, for $n \leq m$, $p$ being a given prime number. The topological group $\mathbf{T}_p = \lim_{\leftarrow} G_n$ is called the $p$-adic solenoid; it is a compact connected commutative group.

a) For every $n$, the continuous homomorphism $f_n: \mathbf{T}_p \to G_n$ is surjective and its kernel is isomorphic to the group $\mathbf{Z}_p$ of $p$-adic integers (cf. III, p. 85, exerc. 1 a)).

b) Let $\varphi$ be the canonical homomorphism $\mathbf{R} \to \mathbf{R}/\mathbf{Z} = \mathbf{T}$. For every $x \in \mathbf{R}$, put $\theta(x) = (\varphi(x/p^n))_{n \in \mathbf{N}}$; now prove that $\theta$ is a continuous injective homomorphism of $\mathbf{R}$ into $\mathbf{T}_p$, and that $\theta(\mathbf{R})$ is a everywhere dense subgroup of $\mathbf{T}_p$.

c) Let I be an open interval in $\mathbf{R}$, of center 0 and of length < 1 ; now prove that in $T_p$, the subspace $f_0^{-1}(\varphi(I))$ is homeomorphic to the product $I \times \mathbf{Z}_p$. In particular, the group $T_p$ is not locally connected. d) Now prove that every closed subgroup H of $T_p$ distinct from $T_p$ and from $\{0\}$, is totally discontinuous and isomorphic to a group of the form $\mathbf{Z}/n\mathbf{Z}$ or $(\mathbf{Z}/n\mathbf{Z}) \times \mathbf{Z}_p$, where n is an integer prime to $p$ (use the prop. 3 of III, p. 62). e) Now prove that $T_p$ is a compact connected indecomposable space, that is to say that there does not exist a covering of $T_p$ formed by two compact connected sets P, Q distinct from $T_p$. (Observe that there exists an integer n such that $f_n(P) \neq G_n$ and $f_n(Q) \neq G_n$, and examine the sets $f_{n+1}(P)$ and $f_{n+1}(Q)$ to obtain a contradiction.)*