  In this paper, we prove that the set of all doubling metrics or all uniformly
disconnected metrics are dense in spaces of metrics on finite-dimensional or
zero-dimensional compact metrizable spaces, and these denseness in spaces of
metrics characterize the finite-dimensionality or zero-dimensionality and the
compactness of metrics spaces. We also determine the topological distribution
of the set of all uniformly perfect metrics in the space of metrics on the
Cantor set.
