  In spaces of metrics, we investigate topological distributions of the
doubling property, the uniform disconnectedness, and the uniform perfectness,
which are the quasi-symmetrically invariant properties appearing in the
David--Semmes theorem. We show that the set of all doubling metrics and the set
of all uniformly disconnected metrics are dense in spaces of metrics on
finite-dimensional and zero-dimensional compact metrizable spaces,
respectively. Conversely, this denseness of the sets implies the
finite-dimensionality, zero-dimensionality, and the compactness of metrizable
spaces. We also determine the topological distribution of the set of all
uniformly perfect metrics in the space of metrics on the Cantor set.
