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 $\leq |2^\omega|$.