Exercise 63, § 1

Let $G$ be the subgroup of $\mathbf{T}^{\mathbf{N}}$ formed by the elements $x= (x_n)$ such that $2x_n=$ 0 for every $n\in \mathbf{N}$ except for at most a finite number.

a) For every finite set $\Lambda \subset \mathbf{N}$, let $U_{\Lambda}\subset G$ be the set of $(x_n)$ such that $x_n= 1$ for $n\in \Lambda$ and $2x_n= 0$ for $n \notin \Lambda$. There exists a topological group structure on G such that the family of sets $U_{\Lambda}$ is a basis of open neighbourhoods of $e$.

b) The topological group G is locally compact.

c) The group G is not divisible.

d) The dual group $\widehat{G}$ is without torsion.