In a $T_3$-space with $\sigma$-locally finite base, every open set is an $F_\sigma$ set.
Solution 1:
For every $x ∈ G$ we have $x ∈ B_{n(x), λ(x)} ⊆ \overline{B_{n(x), λ(x)}} ⊆ G$, so $G = ⋃_{x ∈ G} \overline{B_{n(x), λ(x)}}$, which is not a priori a countable union, but for every $k ∈ ℕ$ we may group the points $x$ and the corresponding sets with $n(x) = k$ together.