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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the dimension of product spaces and an example of M. Wage


Author: Teodor C. Przymusiński
Journal: Proc. Amer. Math. Soc. 76 (1979), 315-321
MSC: Primary 54F45; Secondary 54G20
MathSciNet review: 537097
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Abstract: Modifying a recent example obtained under the assumption of the Continuum Hypothesis by Michael Wage, we prove, without any set-theoretic assumptions beyond ZFC, that for every natural number n there exists a separable and first countable space X such that:

(a) $ {X^n}$ is Lindelöf and $ \dim {X^n} = 0$;

(b) $ {X^{n + 1}}$ is normal but $ \dim {X^{n + 1}} > 0$.

We obtain from this the following corollary. There exists a separable and first countable Lindelöf space X such that:

(a) $ \dim X = 0$;

(b) $ {X^2}$ is normal but $ \dim {X^2} > 0$.

The space X instead of being Lindelöf can be made locally compact and locally countable.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0537097-3
PII: S 0002-9939(1979)0537097-3
Keywords: Strongly zerodimensional spaces, product spaces, normality, Lindelöf property
Article copyright: © Copyright 1979 American Mathematical Society