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## Topology Q+A Board

From: sTudent
Date: July 30, 2006
Subject: Re: omega_1 x (beta N) is not normal

In reply to "Normality, beta N and products", posted by David R. MacIver on July 22, 2006:
>Are there any (nice?) examples of a normal topological space X such that X x beta N is not normal? e.g. does omega_1 work?
>
>Such an example of course has to fail to be paracompact, and I've not really got to grips with many examples of normal spaces which aren't paracompact.
>
>David

Yes, omega_1 works. First omega_1 x (omega_1 + 1) is not normal.
This follows from Tamano's theorem, but it can be proved directly:
in the product the top line and the diagonal can not be separated.
Next, as omega_1 + 1 is a closed subset of 2^{omega_1), we get
that omega_1 x 2^{omega_1) is not normal. Finally, 2^{omega_1) is
separable so it is the perfect image of beta N. The product of two
perfect maps is perfect and a perfect image of a normal space is
normal; all these facts imply that omega_1 x (beta N) is not normal.