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 $A$,


1. there exists a countable collection $\{U\_n:n\in\mathbb N\}$ such that for each $n$, $A \subset U\_n$.
2. $A = \bigcap\_{n\in\mathbb N}\overline{U\_n}$.


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 $G\_\delta$.



Does regular $G_\delta$ imply normal?
A well-known characterisation of normality is useful here (proposition 1.5.15 in Engelking's *General Topology*), I wrote its proof [here](http://at.yorku.ca/p/a/c/a/07.pdf):



> 
> $X$ is normal iff for each closed set $A$ of $X$ and each open set $O$ with $F \subseteq O$, there are open sets $W\_n$, $n \in \mathbb{N}$ of $X$ such that $F \subseteq \bigcup\_n W\_n$ and for all $n$, $\overline{W\_n} \subseteq O$.
> 
> 
> 


The condition of regular $G\_\delta$ dualised to open sets says that every open set of $X$ is the countable union of closed sets whose interiors also cover $O$.
Using these interiors as the $W\_n$ for an open $O$, I think shows that regular-$G\_\delta$ spaces are normal.


