# Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.

$X$ has a regular $G_\delta$-diagonal iff there is a collection of open sets of $X^2$, say $\{U_n: n\in N\}$, such that $\Delta=\bigcap\{\overline{U_n}: n \in N\}$, where $\Delta=\{(x,x): x \in X\}$.

Note that If $X$ is submetrizable then $X$ always has a regular $G_\delta$-diagonal. I know some counterexamples which witnesses the space maybe cannot submetrizable although it has a regular $G_\delta$-diagonal. You can refer Aleksander V. Arhangel’skii & Raushan Z. Buzyakova, The rank of the diagonal and submetrizability, Commentationes Mathematicae Universitatis Carolinae, Vol. 47 (2006), No. 4, 585-597, which is available here in PDF.

These counterexamples are all Tychonoff and it seems difficult to construct.

If we weaken the separate axiom ( The spaces are not needed to Be Tychonoff), then are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Could you construct such counterexamples? The more the better.

Any help will be appreciated.