Characterization of sets in $\mathbb{R}$ that are $F_{\sigma}$ and $G_{\delta}$ I'm looking for a characterization of sets in $\mathbb{R}$ that are both $F_{\sigma}$ and $G_{\delta}$.
Is there anything that must be true for these sets with this  property? For example, must they be dense, or measurable, or something like that. I'm just trying to get an idea of how these specific sets behave.
 A: They don't have to be dense: singletons are both $F_\sigma$ and $G_\delta$, but are not dense (and there are many other examples). They do have to be measurable; in fact, every set which is either $F_\sigma$ or $G_\delta$ (even if it is not both) is measurable, as a countable union or a countable intersection (respectively) of measurable sets (since both closed and open sets are always measurable).
A: The other answer responded to your specific questions about denseness and measurability.  I wanted to point out that every open set is both $F_\sigma$ and $G_\delta$, as is every closed set.  That already gives you a rather large supply of significantly different sets, showing that these sets don't have to have a great deal in common, and in particular topological properties are very unlikely to be guaranteed to hold.
For example, suppose that $A$ is a set which is both $F_\sigma$ and $G_\delta$.  Then:

*

*$A$ might be dense, or might not

*$A$ might be discrete, or might not

*$A$ might have isolated points, or might not

*$A$ might be finite, or countably infinite, or uncountable

*etc.

