Small inductive dimension of completions of metric spaces

We construct a 0-dimensional metric space which under a special set-theoretic assumption, denoted in the paper as S($\aleph _{0}$), does not have a 0-dimensional completion. Shortly after the submission of the paper for publication R. Dougherty has shown the consistency of S($\aleph _{0}$). (S($\aleph _{0}$) disagrees with the continuum hypothesis.)