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!