An example in the dimension theory of metrizable spaces (Q752536)

The author constructs an example of a gap metric space, a metrizable space Z with small inductive dimension \(ind(Z)=0\) and large inductive dimension \(Ind(Z)=1\). Although the first example of a gap metric space was announced in 1962 by P. Roy, such examples remain very scarce in the literature and are among the most complicated of all examples in topology. Using only ZFC, the author constructs his example Z as a dense \(G_{\alpha}\) subspace of \(^{\omega}\omega_ 1\) so that the weight of Z is \(\aleph_ 1\), thus answering a question of Kunen and van Douwen. A. Ostaszewski has independently given an example answering this question.