A note on the Prabir Roy space (Q756758)

Avoiding the continuum hypothesis, the author constructs a complete metric space with a basis of cardinality \(\aleph_ 1\), small inductive dimension \(ind(P)=0\) and large inductive dimension \(Ind(P)=1\). In so doing the author answers a question of \textit{Kunen} and \textit{van Douwen} and provides a slight generalization which includes a similar example due to \textit{J. Kulesza}. The first example of a metric space whose small and large inductive dimensions differ was announced in 1962 by \textit{P. Roy}, however such examples remain very scarce in the literature and are among the most complicated of all examples in topology in particular such examples must necessarily be nonseparable. The author's construction and proofs use the notions of fast sequences, large trees, and the Moore metrization theorem.