TITLE: What's the difference between rationals and irrationals - topologically?
QUESTION [31 upvotes]: I know that sets of rational and irrational numbers are quite different. In measure, almost no real number is rational and of course, $\mathrm{card}(\mathbb Q) < \mathrm{card}(\mathbb R \setminus \mathbb Q) $ tells us that there are indeed much "more" irrationals than rational nubers.
On the other side, we can observe the following

between every two different rationals, there are infinitely many irrations
between every two different irrationals, there are infinitely many rationals
both sets are dense in $\mathbb R$, i.e. every real number can be written as a limit of a sequence of both rationals or irrationals
both are disconnected, neither open nor closed 
...

So I wonder, is there any way to distinguish $\mathbb Q$ and $\mathbb R \setminus \mathbb Q$ from a topological perspective as subspaces of $\mathbb R$? Is there any way to explain why the one set is so much bigger by looking at the topology? In what regard are they different?

REPLY [13 votes]: 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).