On Lipschitz maps and dimension

Maria Moszynska and the first author suggested some natural axioms for fractal dimension functions. We discuss the independence of these axioms. In particular, using the Continuum Hypothesis, we associate to each nonempty separable metric space X a non-negative integer d(X) so that the function d is Lipschitz subinvariant, stable under finite unions, d ([0, 1](n)) = n, but still, for some E subset of [0, 1](3) we have d (E) < dim E, where dim E is the topological dimension of E.