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Strongly rigid metrics in spaces of metrics

Yoshito Ishiki

TL;DR

The paper generalizes the classical notion of strong rigidity from individual metric spaces to the entire space of metrics Met(X), showing that for strongly 0-dimensional metrizable spaces X with Card(X) ≤ 𝔠, strongly rigid metrics are dense in Met(X) and, when X is σ-compact, form a dense G_δ subset; moreover, rigid metrics form a comeager subset under appropriate cardinal assumptions. The core methodology combines discrete building blocks on a large discrete base Ω and its Baire space N(Ω) with linearly independent real numbers over ℚ, S-gauge systems, and prefix-encoding embeddings to produce globally defined strongly rigid metrics; these constructions are then transferred to general X via clopen decompositions and amalgamation. The work yields both density and genericity results for strongly rigid metrics and for metrics with no nontrivial self-isometries, and provides a framework for embedding X into a rigid metric space with controlled diameter or completeness. Overall, the results deepen the connection between dimension theory, metric rigidity, and the topology of the space of metrics, offering new tools for analyzing self-symmetries and metric perturbations in metrizable spaces.

Abstract

A metric space is said to be strongly rigid if no positive distance is taken twice by the metric. In 1972, Janos proved that a separable metrizable space has a strongly rigid metric if and only if it is zero-dimensional. In this paper, we shall develop this result for the theory of space of metrics. For a strongly zero-dimensional metrizable space, we prove that the set of all strongly rigid metrics is dense in the space of metics. Moreover, if the space is the union of countable compact subspaces, then that set is comeager. As its consequence, we show that for a strongly zero-dimensional metrizable space, the set of all metrics possessing no nontrivial (bijective) self-isometry is comeager in the space of metrics.

Strongly rigid metrics in spaces of metrics

TL;DR

The paper generalizes the classical notion of strong rigidity from individual metric spaces to the entire space of metrics Met(X), showing that for strongly 0-dimensional metrizable spaces X with Card(X) ≤ 𝔠, strongly rigid metrics are dense in Met(X) and, when X is σ-compact, form a dense G_δ subset; moreover, rigid metrics form a comeager subset under appropriate cardinal assumptions. The core methodology combines discrete building blocks on a large discrete base Ω and its Baire space N(Ω) with linearly independent real numbers over ℚ, S-gauge systems, and prefix-encoding embeddings to produce globally defined strongly rigid metrics; these constructions are then transferred to general X via clopen decompositions and amalgamation. The work yields both density and genericity results for strongly rigid metrics and for metrics with no nontrivial self-isometries, and provides a framework for embedding X into a rigid metric space with controlled diameter or completeness. Overall, the results deepen the connection between dimension theory, metric rigidity, and the topology of the space of metrics, offering new tools for analyzing self-symmetries and metric perturbations in metrizable spaces.

Abstract

A metric space is said to be strongly rigid if no positive distance is taken twice by the metric. In 1972, Janos proved that a separable metrizable space has a strongly rigid metric if and only if it is zero-dimensional. In this paper, we shall develop this result for the theory of space of metrics. For a strongly zero-dimensional metrizable space, we prove that the set of all strongly rigid metrics is dense in the space of metics. Moreover, if the space is the union of countable compact subspaces, then that set is comeager. As its consequence, we show that for a strongly zero-dimensional metrizable space, the set of all metrics possessing no nontrivial (bijective) self-isometry is comeager in the space of metrics.
Paper Structure (7 sections, 33 theorems, 43 equations)

This paper contains 7 sections, 33 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. A construction of strongly rigid discrete metrics
  5. A system of linearly independent numbers over $\mathbb{Q}$
  6. A construction of strongly rigid metrics
  7. Proofs of the main results

Key Result

Theorem 1.1

Let $X$ be a strongly $0$-dimensional metrizable space with $\mathop{\mathrm{Card}}\nolimits(X)\le \mathfrak{c}$. Let $\epsilon \in (0, \infty)$ and $d\in \mathop{\mathrm{Met}}\nolimits(X)$. Then there exists $e\in \mathrm{LI}(S)$ such that $\mathop{\mathrm{\mathcal{D}}}\nolimits_{X}(d, e)\le \epsil

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • ...and 74 more