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On page 205 of his Topology textbook, James Munkres made an interesting remark:

It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer is affirmative if one assumes the continuum hypothesis.

That's a reference to this paper by Mary Ellen Rudin. However, both Munkres and Rudin were writing decades ago. So my question is, what is the state of research on this problem?

Has it been proven in $ZFC$, or has it been proven to be independent of $ZFC$, or is it still an open problem whether it's a theorem of $ZFC$ or not?

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    $\begingroup$ This survey by Roitman and Williams, from 2015, said that it was still open whether $(\omega + 1)^\omega$ is normal in the box topology. That may not be quite the same thing, though, since "normal" is not the same thing as "hereditarily normal" (I am not an expert on this). But MathSciNet shows that survey as the most recent citation of Rudin's 1972 paper, and presumably anyone who solved the problem would have cited Rudin. $\endgroup$
    – Nate Eldredge
    Commented Nov 8, 2018 at 22:15
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    $\begingroup$ @NateEldredge: $\mathbb R^\omega$ has a closed subspace homeomorphic to $(\omega+1)^\omega$, so if $\mathbb R^\omega$ is normal then so is $(\omega+1)^\omega$. So that survey paper is good evidence that the problem is still open. $\endgroup$
    – Will Brian
    Commented Nov 9, 2018 at 1:40
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    $\begingroup$ @NateEldredge: For what it's worth, it is known that $\mathbb{R}^\omega$ is not hereditarily normal. $\endgroup$
    – Ramiro de la Vega
    Commented Nov 11, 2018 at 20:11
  • $\begingroup$ @NateEldredge this follows from van Douwen's result that the irrationals times countably many convergent sequences in the box topology is not normal, see here or in Mary Ellen's lectures on set theoretic topology . $\endgroup$
    – Henno Brandsma
    Commented Dec 15, 2018 at 9:38

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I'm transcribing here some things from the comments section, so that this question can be marked as answered:

The problem is still open.

This survey by Roitman and Williams, from 2015, said that it was still open whether $(\omega+1)^\omega$ is normal in the box topology. $\mathbb R^\omega$ has a closed subspace homeomorphic to $(\omega+1)^\omega$, so if $\mathbb R^\omega$ is normal then so is $(\omega+1)^\omega$. MathSciNet shows that survey as the most recent citation of Rudin's 1972 paper, and presumably anyone who solved the problem would have cited Rudin.

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