Separable metric topology refined with an $F_\sigma$-set

Question

Let $(X,\tau)$ be a separable metrizable space. Let $A$ be an $F_\sigma$-subset of $X$. Is the topology generated by $\tau\cup \{A\}$ also metrizable?

Answer 1

No, take $A$ to be the rationals and $\tau$ the standard topology on the reals. It's a standard fact (recall Munkres $K$-topology e.g.) that the topology generated by $\tau \cup \{A\}$ is not even regular, let alone metrisable.