On dense subsets in spaces of metrics

Yoshito Ishiki Colloquium Mathematicum MSC: Primary 54E45; Secondary 30L05. DOI: 10.4064/cm8580-9-2021 Published online: 13 April 2022

Abstract

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.

Authors

  • Yoshito IshikiGraduate School of Pure and Applied Sciences
    University of Tsukuba
    Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan
    Current address:
    Photonics Control Technology Team
    RIKEN Center for Advanced Photonics
    2-1 Hirasawa, Wako, Saitama 351-0198, Japan
    e-mail

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