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Modifying a recent example obtained under the assumption of the Continuum Hypothesis by Michael Wage, we prove, without any settheoretic assumptions beyond ZFC, that for every natural number n there exists a separable and first countable space X such that: (a) Xn is Lindelof and dim Xn = 0; (b) Xn I I is normal but dim Xn > 0. We obtain from this the following corollary. There exists a separable and first countable Lindelof space X such that: (a) dim X = 0; (b) X2 is normal but dim X2 > 0. The space X instead of being Lindelof can be made locally compact and locally countable. Assuming the Continuum Hypothesis (CH) Michael Wage constructed a beautiful example of a strongly zerodimensional space X such that x2 is not strongly zerodimensional, thus giving a consistent negative answer to a long outstanding problem in dimension theory, namely, whether dim X x Y 0. COROLLARY. There exists a separable and first countable Lindelof space X such that: (a) dim X = 0; (b) X2 is normal but dim X2 > 0. Thus our theorem provides a "real" counterexample to the conjecture that dim X x Y < dim X + dim Y in the class of normal spaces. The space X, Received by the editors October 1, 1978. AMS (MOS) subject classifications (1970). Primary 54F45, 54B 10, 54D20.

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