Exercise 6, § 7

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

a) For each $n$, the continuous homomorphism $f_n : T_p \to G_n$ is surjective, and its kernel is isomorphic to the group $\mathbf{Z}_p$ of $p$-adic integers [cf. Exercise 1 a)].

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

c) Let $I$ be an open interval in $\mathbf{R}$, with centre $0$ and length $< 1$. Show that the subspace $f_0^{-1}(\varphi(I))$ of $\mathbf{T}_p$ is homeomorphic to the product $I \times \mathbf{Z}_p$. In particular, the group $\mathbf{T}_p$ is not locally connected.

d) Show that every closed subgroup $H$ of $\mathbf{T}_p$, other than $\mathbf{T}_p$ or $\{0\}$, is totally disconnected 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 Proposition 3 of no. 3).

e) Show that $\mathbf{T}_p$ is an indecomposable compact connected space, i.e. that $\mathbf{T}_p$ cannot be covered by two compact connected sets $P, Q$ neither of which is equal to $\mathbf{T}_p$. (Observe that there is 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 get a contradiction). \*