# On the cardinality of ccc spaces with a $G_\delta$-diagonal

In a recent MO post it was noted that Uspenskij's old example of a Tychonoff ccc space with a $G_\delta$ diagonal and arbitrarily large cardinality is not normal. See:

How could I see quickly that this space is not normal?

Is there a normal ccc space with a $G_\delta$ diagonal and cardinality larger than the continuum?

• Taras Banakh notes in the comments that every ccc submetrizable space has cardinality at most continuum. $X$ is submetrizable if it has a weaker metrizable topology; so if $X$ is ccc, also the weaker metrizable topology must be ccc, and hence separable. Therefore the underlying space $X$ will have cardinality at most continuum. So any example answering my question will be a normal ccc space with a $G_\delta$ diagonal which is not submetrizable. But I must confess I don't even know a (non-ccc) example of a normal space with a $G_\delta$-diagonal which is not submetrizable.
• Taras further notes that an example answering my question can't be paracompact. This is because every paracompact space with a $G_\delta$ diagonal is submetrizable.
• Buzyakova proved that every ccc space with a regular $G_\delta$ diagonal has cardinality at most continuum (regular $G_\delta$-diagonal means that there are countable many open neighbourhoods of the diagonal in the square such that the diagonal is equal to the intersection of their closures. Note that every submetrizable space has this property).
• Uspenskij's example is $\sigma$-closed discrete, so in particular it has to contain large closed discrete sets. This is not by chance, as every space with a $G_\delta$-diagonal whose closed discrete sets are countable has cardinality at most continuum (Ginsburg and Woods).
• A subtle quesion, since if instead of $G_\delta$-diagonal take submetrizable, then the answer will be negative. Also a paracompact spaces with $G_\delta$-diagonal is submetrizable and if has ccc, then admits a continuous injective map to a separable metrizable space, so has cardinality of continuum. – Taras Banakh Dec 8 '17 at 18:12
• Even more than that is true: Raushan Buzyakova proved that every ccc space with a regular $G_\delta$ diagonal has cardinality at most continuum and every submetrizable space has a regular $G_\delta$ diagonal. – Santi Spadaro Dec 9 '17 at 9:56
• @SantiSparado Thank for info. Then this is a "grave" question. Or maybe the argument of Buziakova works for $G_\delta$-diagonal? – Taras Banakh Dec 9 '17 at 15:37
• Her argument seems to use the "regularity" of the diagonal essentially. I think that if it's true for normal spaces, a completely different approach is needed. By the way, I realized I don't even know an example of a normal space with a $G_\delta$-diagonal which is not submetrizable, right, not even a non-ccc one... Do you know any, Taras? – Santi Spadaro Dec 12 '17 at 21:01