Can anyone give an example of a Meagre subset of $\mathbb{R}$ (i.e., of the first Baire Category), which isn't an $F_\sigma$?
Simple cardinality arguments show that such a thing exists, but I can't really think of one. Maybe it's impossible to explicitly construct one? A proof (not using cardinalities) would be welcome too (Axiom of Choice, perhaps?).
Take the Cantor set $C$. It is closed and nowhere dense in $\mathbb R$, so $C$ and all its subsets are meager in $\mathbb R$. Now take a countable set $T$ dense in $C$, and consider $A = C \setminus T$. Then $A$ is meager in $\mathbb R$, but is not an $F_\sigma$.