In a 1990 paper*, M.E. Rudin writes (p.137),

So far as is known, normal manifolds may have to be collectionwise Hausdorff [cwH].

Since it holds whenever $V=L$, I understand that at that time, no consistent example of a non-cwH normal manifold was known, but there was no proof that none exist in $\mathsf{ZFC}$.

My question is whether there has been some progress on this question since then. If no definitive answer is known, I'd be glad to know in which models of set theory normal manifolds are cwH.

By a manifold I mean a connected Hausdorff space each of whose points has a neighborhood homeomorphic to $\mathbb{R}^n$. A space is cwH iff given a closed discrete subspace $D$, there is a pairwise disjoint collection of open sets $\{U_d:d\in D\}$ such that $U_d\cap D=\{d\}$.

* Mary Ellen Rudin, Two nonmetrizable manifolds, Topology Appl. 35 (1990), no. 2-3, 137--152, MR1058794.

  • $\begingroup$ Thanks for your edit, @arjafi. I must add that since yesterday, I saw that this problem appears in many papers by Nyikos, in particular in the one linked below he writes that it "has long been one of Rudin’s favorites". people.math.sc.edu/nyikos/Manifolds.pdf $\endgroup$ – Mathieu Baillif Oct 6 '16 at 18:43

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