Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
    
A Space with G-delta Diagonal that is not Submetrizable at Dan Ma's Topology Blog. –  Martin Sleziak Apr 26 '13 at 15:55

1 Answer 1

up vote 7 down vote accepted

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).

share|improve this answer
    
Thanks! 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? –  Paul Dec 28 '11 at 12:16
    
corollary 2.3 of that paper has a necessary and sufficient condition for star-Lindelöf spaces (weakly normal and rank for the diagonal 5), which thus also applies to separable spaces. $\Psi$ is not submetrizable because a submetrizable pseudocompact space is metrizable (also in the paper). –  Henno Brandsma Dec 28 '11 at 14:07
    
@John: I can’t add much to what Henno has already mentioned. One other line of attack is via Dieudonné completeness, since every submetrizable Tikhonov is Dieudonné complete; the concept is defined and numerous basic results about it are given in Exercise 8.5.13 of Engelking’s General Topology. –  Brian M. Scott Dec 28 '11 at 23:37
    
@BrianM.Scott,the answer and the remarks feed me a lot. Thanks! –  Paul Dec 29 '11 at 0:04

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.