I'm trying to prove that if every closed set in a topological space is regular $G_\delta$, then the space is normal. By *regular $G_\delta$*, I mean for any closed set $A$,

- there exists a countable collection $\{U_n:n\in\mathbb N\}$ such that for each $n$, $A \subset U_n$.
- $A = \bigcap_{n\in\mathbb N}\overline{U_n}$.

When I try to prove this, I encounter "a countable intersection of nested nonempty closed sets" and I require this to be nonempty for my proof to work. Of course, this is not true in general.

I've also looked for counter examples - The Moore plane is almost a counter example, since it is not normal but every closed set is $G_\delta$. However, I haven't been able to show that every closed set is regular $G_\delta$.