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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 $G_\delta$.

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A well-known characterisation of normality is useful here (proposition 1.5.15 in Engelking's General Topology), I wrote its proof here:

$X$ 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.

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