How does one show that the set of rationals is topologically disconnected? Let $\mathbb{Q}$ be the set of rationals with its usual topology based on distance:
$$d(x,y) = |x-y|$$
Suppose we can only use axioms about $\mathbb{Q}$ (and no axiom about $\mathbb{R}$, the set of reals). Then how can we show that $\mathbb{Q}$ is topologically disconnected, i.e.: there exist two open sets $X$ and $Y$ whose union is $\mathbb{Q}$?
If we were allowed to use axioms about $\mathbb{R}$, then we could show that for any irrational number $a$:


*

*if $M$ is the intersection of $]-\infty, a[$ with the rationals, then $M$ is an open set of $\mathbb{Q}$

*if N is the intersection of $]a, +\infty[$ with the rationals, then $N$ is an open set of $\mathbb{Q}$

*$\mathbb{Q}$ is the union of $M$ and $N$. CQFD. 


But if we are not allowed to use axioms about $\mathbb{R}$, just axioms about $\mathbb{Q}$?

The rationals is the union of two disjoint open sets $\{x\in\mathbb{Q}:x^2>2\}$ and $\{x\in\mathbb{Q}:x^2<2\}$.
