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If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?

Partial answers, which I proved in the 1980's, include:

(0) The answer is consistently yes if the space has cardinality $\aleph_1$.

(1) Yes under MA, if the cardinality of the space is less than $\mathfrak{c}$.

(2) Yes, if the space is countably metacompact.

(3) Such a space must be collectionwise Hausdorff.

(4) It is not known if such a space is collectionwise normal (at least not to me).

(5) Any counterexample would be a Dowker space with a $\sigma$-disjoint base.

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    $\begingroup$ Have you asked the question you intended? You say it is normal at the beginning. $\endgroup$
    – Joel David Hamkins
    Commented Sep 13, 2023 at 11:49
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    $\begingroup$ Union of open metrizable subspaces implies first-countable, hence the latter condition is redundant. $\endgroup$
    – Emil Jeřábek
    Commented Sep 13, 2023 at 12:10
  • $\begingroup$ This is a very interesting problem, and it is still open, as far as I know. Peter Nyikos was working on it a few years ago and I heard him lecture about it a few times. I don't think he made much progress beyond what's already mentioned in Mike's answer, though. $\endgroup$
    – Will Brian
    Commented Sep 13, 2023 at 15:36
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    $\begingroup$ Hi Mike, welcome to MO. If you'd like to clarify your question further, or include some more of what you know about it, then the normal way to do that is to edit the question and include that information as part of the question. I hope you don't mind -- I'm going to edit your question and move the data from the answer you posted so as to make it part of the question. $\endgroup$
    – Will Brian
    Commented Sep 13, 2023 at 15:38
  • $\begingroup$ Apparently Peter wrote down some of his thoughts on this problem: people.math.sc.edu/nyikos/Reed.pdf $\endgroup$
    – Will Brian
    Commented Sep 13, 2023 at 15:43

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