Example

We may define a topology on the set $Q$ of rational numbers by taking for open sets all unions of bounded open intervals; the set of these subsets certainly satisfies (O_I), and to see that it satisfies (O_II) it is enough to remark that if the intersection of two open intervals $]a, b[$ and $]c, d[$ is not empty, then it is the interval $]\alpha, \beta[$, where $\alpha = \sup(a, c)$ and $\beta = \inf(b, d)$. We get the same topology by defining for each $x \in Q$ the set $\mathfrak{B}(x)$ of neighbourhoods of $x$ to be the set of subsets containing an open interval to which $x$ belongs. The topological space obtained by assigning this topology to $Q$ is called the rational line (cf. Chapter IV, § 1, no. 2). Notice that in this space every open interval is an open set. \* We can define a topology on the set $R$ of real numbers in the same way; $R$ with this topology is called the real line (cf. § 2, Exercise 5 and Chapter IV, § 1, no. 3). \*