One can show that the group $\text{Ext}^1(\mathbf Q, \mathbf Z)$ (calculated in $Ab$) identifies naturally with $\mathbf A_f/\mathbf Q$, where $\mathbf A_f$ is the additive group of finite adèles. More precisely, the long exact sequence of $\mathbf Z \to \mathbf Q \to \mathbf Q/\mathbf Z$ identifies it with $\mathbf A_f/\mathbf Q$. The group $\mathbf A_f/\mathbf Q$ has a nice topology coming from the fact that $\mathbf Q$ is a discrete subgroup of $\mathbf A_f$. It is a locally compact group.

I am wondering if there is any way to define the topology on $\text{Ext}^1(\mathbf Q, \mathbf Z)$ intrinsically, without identifying it with $\mathbf A_f/\mathbf Q$, or even without looking at any particular exact sequence...