On dense subsets in spaces of metrics

TL;DR

This paper studies the distribution of certain geometric properties in metric spaces, showing how their density relates to the underlying space's dimensionality and characterizing their distribution on specific spaces like the Cantor set.

Contribution

It establishes the density of doubling and uniformly disconnected metrics in relevant spaces and links these properties to the space's dimensionality and compactness.

Findings

Doubling metrics are dense in spaces of metrics on finite-dimensional spaces.

Uniformly disconnected metrics are dense on zero-dimensional compact spaces.

The distribution of uniformly perfect metrics is characterized on the Cantor set.

Abstract

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.