Adele ring

Remarks

With this topology, $\mathbb{A}_K$ is a locally compact topological ring. The topology is finer than the one induced by the product topology on $\prod_v K_v$. The diagonal map $K \to \mathbb{A}_K$, $x \mapsto (x)_v$, is well defined because an element of $K$ lies in $\mathcal{O}_v$ for all but finitely many $v$; its image is discrete and $\mathbb{A}_K / K$ is compact.

Example: for $K = \Q$, the places are $\infty$ and the primes $p$, with $\Q_\infty = \R$, $\Q_p$ the p-adic numbers and $\mathcal{O}_p = \Z_p$ the p-adic integers. The family $x = (x_v)$ with $x_\infty = \pi$, $x_2 = 1/2$ and $x_p = 1$ for $p \ge 3$ is an adele, since only one $p$-adic entry lies outside $\Z_p$.

Non-example: the family with $x_\infty = 0$ and $x_p = 1/p$ for every prime $p$ lies in $\prod_v \Q_v$ but not in $\mathbb{A}_{\Q}$, since $1/p \notin \Z_p$ for infinitely many $p$.