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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!

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    [A Space with G-delta Diagonal that is not Submetrizable](http://dantopology.wordpress.com/2012/08/10/a-space-with-g-delta-diagonal-that-is-not-submetrizable/) at Dan Ma's Topology Blog.2013-04-26

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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).

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    @BrianM.Scott,the answer and the remarks feed me a lot. Thanks!2011-12-29