Are regular Hausdorff $G_\delta$ spaces normal?

Question

A $G_\delta$ space is a space in which every closed set is a $G_\delta$ set. Combining $G_\delta$ space with normal is kind of a strong condition, as it gives the class of perfectly normal spaces. I am wondering what happens is normal is weakened to regular.

Is there an example of a regular Hausdorff $G_\delta$ space that is not normal?

Or same question for Tychonoff $G_\delta$ spaces.

Answer 1

The Moore plane is an example. It is Tychonoff and closed subsets are $G_\delta$, but it is not normal.

That this space is Tychonoff and not normal is mentioned in the wiki article linked above. For the fact that it has closed sets $G_\delta$, see this post: Moore plane / Niemytzki plane and the closed $G_\delta$ subspaces.

Answer 2

The Sorgenfrey plane is another example. It has a base of clopen sets, so it is Tikhonov, and it is a well-known example of a non-normal product of normal spaces. Finally, Dan Ma’s Topology Blog has a proof that it is a $G_\delta$-space.