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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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    Thank you, Gerry Myerson, and t.b.2011-10-22

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