# Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

• separable = $X$ has a countable dense subset.
• A space $X$ has a zeroset-diagonal when there is a continuous function $f:X^2 \rightarrow [0,1]$ with $\Delta=f^{-1}(0)$ where $\Delta=\{(x,x)\mid x\in X\}$ is the diagonal.
• CCC = countable chain condition = every family of disjoint nonempty open sets is countable.
• Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.

If this example exists, the cardinality of $X$ must be $\leq |2^\omega|$, for Buzyakowa has proved if $X$ has ccc and regular $G_\delta$-diagonal (weaker than zeroset-diagonal) then the cardinality must be $\leq |2^\omega|$.

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Hi, John, welcome to math.SE! I made some changes in your formatting because that looks more like what you intended. Would you mind changing the title into something more descriptive, e.g. the first line in this post? –  t.b. Oct 21 '11 at 7:31
Thanks. I'm very appreciate what you do for me. –  Paul Oct 21 '11 at 8:47
I edited the title. –  Gerry Myerson Oct 21 '11 at 10:26
For the sake of transparency: crossposted to MO –  t.b. Oct 21 '11 at 11:05
Thank you, Gerry Myerson, and t.b. –  Paul Oct 22 '11 at 1:05