Definition 1

The rational line is the topological space consisting of the set $\mathbf{Q}$ together with the additive group topology for which the symmetric open intervals $]-a, +a[$ $(a > 0)$ form a fundamental system of neighbourhoods of $0$.

The topological group $\mathbf{Q}$ thus defined is called the additive group of the rational line.

If $a$ is any rational number $> 0$, there is an integer $n > 0$ such that $1/n < a$; hence the open intervals $]-\frac{1}{n}, +\frac{1}{n}[$ $(n = 1, 2, \ldots)$ form a fundamental system of neighbourhoods of $0$ on the rational line.

We obtain a fundamental system of neighbourhoods of any point $x \in \mathbf{Q}$ by taking the open intervals $]x - a, x + a[$, where $a$ runs through the set of rational numbers $> 0$ (or the set of numbers $1/n$).

Definition 1 is therefore equivalent to that given in Chapter I, § 1, no. 2.

For each pair of rational numbers $(a, b)$ such that $a < b$, there exists $c \in \mathbf{Q}$ such that $a < c < b$ [for example $c = (a + b)/2$]; it follows that the rational line is a non-discrete Hausdorff space.

For each $a > 0$, let $U_a$ be the set of pairs $(x, y)$ in $\mathbf{Q} \times \mathbf{Q}$ such that $|x - y| < a$. As $a$ runs through the set of rational numbers $> 0$ (or just the set of numbers $1/n$), the sets $U_a$ form a fundamental system of entourages of the uniformity of the additive group $\mathbf{Q}$ of the rational line. Relations (2) and (3) show that $|x|, x^+$ and $x^-$ are uniformly continuous on $\mathbf{Q}$. It follows that the functions $\sup(x, y)$ and $\inf(x, y)$ are uniformly continuous on $\mathbf{Q} \times \mathbf{Q}$.