TITLE: $\mathbb{Q}$ is not open, is not closed, but is the countable union of closed sets.
QUESTION [7 upvotes]: I want to prove that $\mathbb{Q}$ (the set of rational numbers) is not open, is not closed, but is the countable union of closed sets. I tried to show that $\mathbb{Q}$ doesn't contain all of its limit points which would imply not closed. However, I am not able to prove the other two things. I need little help to prove this. Thanks.

REPLY [7 votes]: $\Bbb Q$ is not closed because it is dense, and if a set is both dense and closed then it is equal to the whole space (in this case, $\Bbb R$).
$\Bbb Q$ is not open either because open sets are either empty, or contain an interval which makes them uncountable; but $\Bbb Q$ is countably infinite so it is neither empty nor uncountable.
Lastly $\Bbb Q$ is the countable union of singletons, all of which are closed.
Interestingly enough, $\Bbb Q$ cannot be written as an intersection of countably many open sets.