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

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.