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can someone tell me where can i find a proof of the following theorem (by A.H.Stone) : "an uncountable product of Hausdorff non-compact spaces is never normal " ?

thanks in advance !

thetruth
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1 Answers1

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You cannot find such a proof, as it is false. E.g. ${\omega_2}^{\omega_1}$, where ordinals have the order topology, is a normal space.

What A.H. Stone did prove, e.g. it follows directly from theorem 4 in this paper from 1948, was that no uncountable product of non-compact metric spaces is normal. This follows quite directly from the fact that $\omega^{\omega_1}$ is not normal (in that same paper) plus the fact that for metric spaces being non-compact implies that it contains a closed copy of a countable discrete set (so $\omega$). See this related thread for proofs.

In fact this can be generalized to paracompact $p$-spaces etc. More info on this is in the excellent survey paper on products of normal spaces, in the Handbook of Set Theoretic Topology. There you can also find the proof of the normality of my counterexample.

Community
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Henno Brandsma
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    Thanks for this answer. I should have checked in the Handbook. I edited the Wikipedia page on normal spaces accordingly (it contained the statement in the question and attributed it to Stone). – t.b. Sep 10 '12 at 11:01
  • thank you Henno for the wonderful answer and thank you t.b. for editing the wikipedia page (in fact, i saw this pseudo-theorem there). – thetruth Sep 10 '12 at 11:47