Recently, I'm interested in the topological spaces with $G_\delta$ diagonal. Could someone give me some examples such that the given topology space is a Tychonoff space with a $G_\delta$ diagonal but not submetrizable? The more, the better.
Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.
One more question: How could I know the topological space is not submetrizable? Such as $X$ is not $T_2$ or has not $G_\delta$ diagonal, or has not regular $G_\delta$ diagonal, or has not zeroset diagonal. Is there some other tools that I could use to judge that the space is not submetrizable?
Thanks in advance!
You probably want to look at 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. Example 2.9 is a separable Tikhonov Moore space that is not submetrizable but has a $G_\delta$-diagonal. (In fact it has a rank $3$ diagonal.) Example 2.17, due to Mike Reed, is a non-separable Tikhonov Moore space that is not submetrizable but has a $G_\delta$-diagonal.
Another example is the Mrówka space $\Psi$, which is Tikhonov, separable, pseudocompact, not countably compact, and not submetrizable but does have a $G_\delta$-diagonal (even a rank $2$ diagonal).