# Is ω1 × βN normal?

Once upon a time I asked whether $\omega_1 \times \beta \mathbb{N}$ is normal. I got the answer no and a fairly convincing proof of this here

However I'm currently in a situation where I have three plausible proofs of plausible results at most two of which can be true, and of the three this is the one of which I'm currently the least sure (there are a bunch of details I haven't yet fully checked as I'm in the process of dusting off some of my recently rather unused knowledge about general topology), so was hoping someone could confirm.

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I fixed the link. –  Joel David Hamkins Apr 3 '10 at 12:01
Thanks. Sorry about that. My MathOverflow-fu is still rather white belt. –  David R. MacIver Apr 3 '10 at 12:16
What is $\omega_1$ and $\beta N$? –  Thomas Kragh Apr 3 '10 at 13:41
$\omega_1$ is the first uncountable ordinal number with the order topology. $\beta N$ should be $\beta \mathbb{N}$ (I'll fix that) and is the Stone-Cech compactification of the natural numbers with the discrete topology. –  David R. MacIver Apr 3 '10 at 14:01
By the way, I've proven one of the other two results false (or at least, the proof is wrong), so my belief in this one is reaffirmed. It would be good to have something to reference in a bibliography though if anyone can provide one. –  David R. MacIver Apr 3 '10 at 14:02

## 1 Answer

I have reread the proof and it's completely correct. The idea is that $\omega_1 \times \beta\mathbb{N}$ maps perfectly onto a non-normal space, and normality is preserved under perfect maps. Tamano's theorem says that $X$ is paracompact Hausdorff iff $X \times \beta X$ is normal, and we use that $\omega_1$ is not paracompact and $\beta \omega_1 = \omega_1 + 1$. But the direct proof as sketched is also correct (we can use the pushdown lemma to prove it).

All building blocks can be found in Engelking, e.g.

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Thanks for double checking. As per comment, my reason for doubt had already evaporated. I'd also verified the steps myself from engelking. Now the only question is how to cite this. :-) –  David R. MacIver Apr 4 '10 at 9:39