On dense subsets in spaces of metrics
In spaces of metrics, we investigate topological distributions of the doubling property, uniform disconnectedness, and uniform perfectness, which are 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 implies the finite-dimensionality, zero-dimensionality, and 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.