Vtyjkyuk

Let $X$ be a topological space.

1. We say it is normal if for any two disjoint closed sets $A$ and $B$, we can find open sets $U$ and $V$ such that $A subset U$, $B subset V$, and $U cap V = varnothing$.




2. We say $X$ is regular $G_delta$, if for any closed set $A$ in $X$, we can find a countable collection of open sets $U_n: n in mathbb N$ such that $$A = bigcap_nin mathbb N overline U_n,quad mathrmandquad forall n in mathbb N: A subset U_n.$$

I'm trying to prove that if $X$ is regular $G_delta$, then $X$ is normal. Here is my attempted proof outline (which doesn't work!)

Let $A$ and $B$ be disjoint closed subsets of $X$. Then $A$ and $B$ are regular $G_delta$ sets, so let $U_n:nin mathbb N$ be a collection of open sets each containing $A$, the intersection of whose closures is $A$, and $V_m:min mathbb N$ a collection of open sets each containing $B$, the intersection of whose closures is $B$. Without loss of generality we assume each $U_n$ is disjoint from $B$, and each $V_n$ is disjoint from $A$. (We can do this by taking intersections with $Xsetminus B$ and $Xsetminus A$ respectively.) Since $A$ and $B$ are disjoint, there must exist $n in mathbb N$ such that $overline U_n cap B = varnothing$. Similarly, pick $minmathbb N$ such that $overline V_m cap A = varnothing$. Then $U_nsetminus overlineV_m$ is an open set containing $A$, and $V_m setminus overline U_n$ is an open set containing $B$, whose intersection is empty. Thus $X$ is normal.

The reason it doesn't work is because the existence of $n$ such that $overline U_n cap B = varnothing$ is not true - If I try to prove by contradiction that such an $n$ exists, I will need to prove that a countable intersection of nested non-empty closed sets is non-empty, but this is not true in general.

I've also attempted to disprove this. The Moore plane is an example of a non-normal topological space such that every closed set is $G_delta$. However, I was unable to show that every closed set is regular $G_delta$.

Could someone provide some guidance? Thank you

For future reference - if anybody has the same question:
I posted the question to MathOverflow and got an answer https://mathoverflow.net/questions/309139/does-regular-g-delta-imply-normal

The crux is: use the characterisation

$X$ is normal iff for each closed set $F$ of $X$ and each open set $O$ with $F subseteq O$, there are open sets $W_n$, $n in mathbbN$ of $X$ such that $F subseteq bigcup_n W_n$ and for all $n$, $overlineW_n subseteq O$.

Then one can answer my question in the affirmative: Regular $G_delta$ implies normal.