$\mathbb{Q}$ closed in the adeles? It is relatively easy to see that $\mathbb{Q}$ (diagonally embedded) is dense in $\mathbb{A}_\mathrm{fin} = \hat{\prod}^{Z_p} Q_p$ (the 'finite adeles where the restricted product is only taken for the finite places $p$) so it cannot be a closed subset. My question is: can one see that the diagonal embedding of whole $\mathbb{Q}$ is not closed in the full adeles $\mathbb{R} \times \mathbb{A}_\mathrm{fin}$? Does somebody know what the closure is?
Thanks in advance,
Fabian Werner

"Recall" the following:


*

*$\mathbb{Q}$ is discrete in $\mathbb{A} = \mathbb{R}\times\mathbb{A}_{\text{fin}}$.

*$\mathbb{A}$ is Hausdorff.

*Every discrete subgroup of a Hausdorff group is closed.

