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