Exercise 26, § 6

a) Let $p, q$ be two distinct prime numbers and let $\mathfrak{T}$ be the topology on $\mathbf{Q}$ which is the least upper bound of the topologies $\mathfrak{T}_p$ and $\mathfrak{T}_q$. Then $\mathfrak{T}$ is locally bounded [Exercise 20 c)] and is compatible with the field structure of $\mathbf{Q}$. With respect to this topology neither of the two sequences $((p/q)^n)_{n \geq 0}$, $((q/p)^n)_{n \geq 0}$ is bounded; the sequence $(1/p^n)_{n \geq 0}$ is not bounded, but the sequence $(p^n)_{n \geq 0}$, which is bounded, does not tend to 0 [cf. Exercise 22 d)]. Show that the completion of $\mathbf{Q}$ with respect to the topology $\mathfrak{T}$ is isomorphic to the product of the topological fields $\mathbf{Q}_p$ and $\mathbf{Q}_q$.

b) If $P$ is the set of all prime numbers then the least upper bound $\mathfrak{T}_0$ of the topologies $\mathfrak{T}_p$ for $p \in P$ is compatible with the field structure of $\mathbf{Q}$ but is not locally bounded [Exercise 20 c)]. What is the completion of $\mathbf{Q}$ with respect to the topology $\mathfrak{T}_0$?