The Complex Stone-Weierstrass Property

Kenneth Kunen

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We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than $c$ has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight $c$ which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of $π$ -weight less than $𝔭$ has a dense completely Hausdorff (and hence Urysohn) subspace....