What's the difference between rationals and irrationals - topologically?

• The irrational numbers are a Baire space (and also the Baire space), and they are completely metrizable. This means that there is a complete metric space which is homeomorphic to the irrational numbers with the subspace topology.

  The rational numbers, on the other hand, are not a Baire space and they are not completely metrizable which means they are not homeomorphic to any complete metric space.

• From a "local" (Borel) perspective the irrationals are a $G_\delta$ set which is not $F_\sigma$, and the rationals (consequentially) are an $F_\sigma$ set which is not $G_\delta$. This means that the irrationals are the intersection of countably many open sets, but not the union of countably many closed sets.

  I should add that being $G_\delta$ is sometimes denoted as $\bf\Pi^0_2$, and being $F_\sigma$ can be denoted as $\bf\Sigma^0_2$.

The two properties are almost the same. It can be shown that $G_\delta$ subsets of a complete metric space (like $\mathbb R$) are exactly those subsets which are completely metrizable. On the other hand, having no isolated points and being completely metrizable means that you're a Baire space (and therefore the rationals are not such space).

Both spaces have classical characterizations: the rationals $\mathbb{Q}$ is the (up to homeomorphism) unique countable metric space without isolated points. The irrationals are the (up to homeomorphism) unique separable metric space that is completely metrizable, zero-dimensional (it has a base of clopen sets) and nowhere locally compact (no point has a compact neighbourhood). All of these properties $\mathbb{Q}$ also has, except being completely metrizable (because it is not a Baire space).