$G_\delta$ sets I understand that a $G_\delta$ set is a set which is a countable intersection of open sets. My question is: Is there any other characterization for $G_\delta$ sets (on $\mathbb{R}$)?
For example, can I say that the interior of this sets is not empty? or that they are dense somewhere (means that they are not nowhere dense)? or any other topological characterization? Somehow I find it difficult to imagine those sets.
Also, How can I prove that the $\mathbb{Q}$ is not a $G_\delta$ set.
Thank you,
Shir
 A: For the question of why $\mathbb{Q}$ is not $G_{\delta}$, you may want to use the following two results:


*

*A non-empty countable complete metric space has an isolation point (a pretty straight forward corollary of Baire category theorem).

*A subspace of a complete metric space is completely metrisable if and only if it is a $G_{\delta}$ subset.
Use $1.$ to infer that $\mathbb{Q}$ is not completely metrisable, and $2.$ to conclude that it thus can not be a $G_{\delta}$ subset of the complete metric space $\mathbb{R}$.
A: In the case of the $\Bbb R$, $G_\delta$ subsets do not have many restrictions in terms of interior or denseness. But you can prove that every closed subset of $\Bbb R$ is also $G_\delta$. An equivalent, perhaps easier, thing to prove is that every open subset of $\Bbb R$ is $F_\sigma$. You can certainly prove that an open interval is $F_\sigma$. If you know in addition that every open subset of $\Bbb R$ is a countable disjoint union of open intervals and that a countable union of $F_\sigma$ subsets is still $F_\sigma$, you can prove that result.
The best way to see that $\Bbb Q$ is not a $G_\delta$ will often use the Baire Category Theorem. If you know that $\Bbb R$ is a Baire space, then you can show $\Bbb Q$ is not $G_\delta$ easily with this result that you should do as an exercise: A dense $G_\delta$ subset in any topological space is comeager. Why does this finish it? Because $\Bbb Q$ is already meager as it is a countable union of singletons. If $\Bbb Q$ were comeager, $\Bbb R$ would be the union of two meager subsets.
