Exercise 9, § 2

a) Let $\boldsymbol{a}= (a_n)_{n\geqslant 0}$ be a sequence of integers $>1$. We endow $\mathbf{Z}$ with the structure of a topological ring for which the subsets $\mathbf{Z}a_0a_1\cdots a_n$, where $n\geqslant 0$, form a fundamental system of neighbourhoods of 0. We denote by $\Delta_{\boldsymbol{a}}$ the completion of this ring. It is compact, metrizable and totally disconnected. If $p$ is a prime number, and if $a_i=p$ for all $i$, we have $\Delta_{\boldsymbol{a}}=\mathbf{Z}_p$.

b) We denote by $\mathbf{Z}(\boldsymbol{a}^{\infty})$ the multiplicative group of complex numbers of the form exp(2$i\pi l/(a_0a_1\cdots a_r)$) where $l\in \mathbf{Z}$ and $r\in \mathbf{N}$, endowed with the discrete topology. Then $\chi \mapsto \chi (1)$ is an isomorphism of $\widehat{\Delta}_{\boldsymbol{a}}$ onto $\mathbf{Z}(\boldsymbol{a}^{\infty})$.

c) Let B be the subgroup of $\mathbf{R}\times \Delta_{\boldsymbol{a}}$ formed by the $(n, n)$ for $n\in \mathbf{Z}$; it is discrete. We put $\Sigma_{\boldsymbol{a}}= (\mathbf{R}\times \Delta_{\boldsymbol{a}})/B$. It is a compact metrizable connected group. (Observe that the canonical image of $\mathbf{R}$ in $\Sigma_{\boldsymbol{a}}$ is dense, and use lemma 1 of II, p. 244.)

d) Let $\Gamma_{\boldsymbol{a}}$ be the additive subgroup of the discrete group $\mathbf{Q}$ formed by the rational numbers of the form $l/(a_0a_1\cdots a_r)$ where $l\in \mathbf{Z}$ and $r\in \mathbf{N}$. If $\chi \in \widehat{\Sigma}_{\boldsymbol{a}}$, then $\chi$ defines a unitary character of $\mathbf{R}\times \Delta_{\boldsymbol{a}}$, hence a unitary character of $\mathbf{R}$ of the form $x\mapsto$ exp(2$i\pi \alpha_{\chi}x$) where $\alpha_{\chi}\in \mathbf{R}$. Then $\chi \mapsto \alpha_{\chi}$ is an isomorphism of $\widehat{\Sigma}_{\boldsymbol{a}}$ onto $\Gamma_{\boldsymbol{a}}$.

e) Now prove that $\widehat{\mathbf{Q}}$ is canonically isomorphic to $\Sigma_{\boldsymbol{a}}$, where the sequence $\boldsymbol{a}$ is defined by $a_n=n+ 2$ for all $n\geqslant 0$.

f) Let $\mathbf{R}_d$ be the group $\mathbf{R}$ endowed with the discrete topology. Now prove that $\widehat{\mathbf{R}}_d$ is isomorphic to $(\widehat{\Sigma}_{\boldsymbol{a}})^{\mathfrak{c}}$ where $a_n=n+ 2$ and where $\mathfrak{c}$ is the power of the continuum. (Consider a basis of $\mathbf{R}$ as a $\mathbf{Q}$-vector space.)