A space has a $G_\delta$-diagonal if its diagonal can be written as the intersection of countably many open subsets of the square. A space is submetrizable if it has a weaker metrizable topology. Every submetrizable space has a $G_\delta$-diagonal, but the converse is not true. The simplest example of a space with a $G_\delta$-diagonal which is not submetrizable is the Mrowka-Isbell space $\Psi(\mathcal{A})$ where $\mathcal{A}$ is a maximal almost disjoint family of subsets of $\omega$ (see this post by Daniel Ma for details).

However $\Psi(\mathcal{A})$ is not normal, for example because $\Psi(\mathcal{A})$ is a pseudocompact space which is not countably compact.

Is there a normal space with a $G_\delta$-diagonal which is not submetrizable?

Strengthening normality to paracompactness we get a positive result (see Gary Gruenhage's chapter in the Handbook of Set-theoretic Topology).

The above question is connected to this other question of mine (see the post for details about the connection):

On the cardinality of ccc spaces with a $G_\delta$-diagonal

The space in that question is required in addition to be ccc, so we may also ask a possibly stronger question:

Is there a normal ccc space with a $G_\delta$-diagonal which is not submetrizable?

Two additional examples of Tychonoff separable spaces with a $G_\delta$-diagonal which are not submetrizable have been constructed by Buzyakova. The main advantage of her examples over the Mrowka-Isbell space is that they have some additional compactness-type properties (for example, they don't contain any uncountable closed discrete sets). However, her spaces are not normal: the second one is not normal because it's a pseudocompact non-countably compact space and she explains why the first one is not normal in the Remark at the end of page 16, adding that she has tried to make it normal.


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