nLab
rational number

The field of rational numbers, denoted $\mathbb{Q}$, is the field of fractions? of the commutative ring of integer?s, $\mathbb{Z}$.

There are several interesting topologies on $\mathbb{Q}$ that make $\mathbb{Q}$ into a topological group under addition, allowing us to define interesting fields by taking the completion with respect to this topology:

1. The discrete topology is the most obvious, which is already complete.
2. The absolute-value topology is defined by the metric $d(x,y)≔\mid x-y\mid$; the completion is the field of real numbers.
3. Fixing a prime number? $p$, the $p$-adic topology is defined by the ultrametric $d(x,y)≔1/n$ where $n$ is the highest exponent on $p$ in $x$ or $y$; the completion is the field of $p$-adic number?s.

Interestingly, (2) cannot be interpreted as a localic group, although the completion $ℝ$ can. (Probably the same holds for (3); I need to check.)