Metrizability of a compact Hausdorff space whose diagonal is a zero set Let $X$ be a compact Hausdorff space and suppose the diagonal in $X\times X$ is the zero set of a nonnegative continuous real-valued function.  Why does this imply $X$ is metrizable?

This should be well-known Šneider theorem asserting that a Hausdorff compact space with $G_\delta$-diagonal is metrizable (see, for instance, Gru, 2.13).
[Gru] Gary Gruenhage. Generalized metric spaces in Handbook of set-theoretic topology, ed. K. Kunen, J. Vaughan, North-Holland, 1984.
