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An interpolation of metrics and spaces of metrics

Yoshito Ishiki

TL;DR

The work develops an interpolation framework for metrics on closed subsets of metrizable spaces, using $\mathcal{D}_{X}$ and Kuratowski embeddings to glue local metrics into global ones. It introduces the notion of transmissible parameters to capture geometric properties and proves that, for singular parameters, the anti-$\mathfrak{G}$-transmissible property occurs generically as a dense $G_{\delta}$ subset of $\mathrm{Met}(X)$ (and $\mathrm{Comp}(X)$) for non-discrete spaces. Consequently, many metric-geometric properties (doubling, uniform disconnectedness, rich pseudo-cones, metric inequalities, etc.) fail generically, while related local versions hold in appropriate spaces. The results yield a broad, structural view of the generic geometry that can exist on moduli spaces of metrics, with implications for typicality in moduli spaces of metrics and the prevalence of complex geometric behavior. The methods integrate interpolation, embedding into Banach spaces, and selection theorems to connect local extensions with global metric structure.

Abstract

As a generalization of Hausdorff's extension theorem of metrics, we prove an interpolation theorem of a family of metrics defined on closed subsets of metrizable spaces. As an application, we investigate typicality of subsets of moduli spaces of metrics. We observe that various sets of all metrics with properties appearing in metric geometry are dense intersections of countable open subsets in spaces of metrics on metrizable spaces. For instance, our study is applicable to the set of all non-doubling metrics and the set of all non-uniformly disconnected metrics.

An interpolation of metrics and spaces of metrics

TL;DR

The work develops an interpolation framework for metrics on closed subsets of metrizable spaces, using and Kuratowski embeddings to glue local metrics into global ones. It introduces the notion of transmissible parameters to capture geometric properties and proves that, for singular parameters, the anti--transmissible property occurs generically as a dense subset of (and ) for non-discrete spaces. Consequently, many metric-geometric properties (doubling, uniform disconnectedness, rich pseudo-cones, metric inequalities, etc.) fail generically, while related local versions hold in appropriate spaces. The results yield a broad, structural view of the generic geometry that can exist on moduli spaces of metrics, with implications for typicality in moduli spaces of metrics and the prevalence of complex geometric behavior. The methods integrate interpolation, embedding into Banach spaces, and selection theorems to connect local extensions with global metric structure.

Abstract

As a generalization of Hausdorff's extension theorem of metrics, we prove an interpolation theorem of a family of metrics defined on closed subsets of metrizable spaces. As an application, we investigate typicality of subsets of moduli spaces of metrics. We observe that various sets of all metrics with properties appearing in metric geometry are dense intersections of countable open subsets in spaces of metrics on metrizable spaces. For instance, our study is applicable to the set of all non-doubling metrics and the set of all non-uniformly disconnected metrics.

Paper Structure

This paper contains 21 sections, 38 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main results
  4. Interpolation of metrics
  5. Moduli spaces of metrics
  6. Organization
  7. Preliminaries
  8. The Hausdorff extension theorem
  9. The Michael continuous selection theorem
  10. The Kuratowski embedding theorem
  11. Baire spaces
  12. An interpolation Theorem of metrics
  13. Amalgamation lemmas
  14. Proof of Theorem \ref{['thm:interpolate']}
  15. Transmissible properties
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X$ be a metrizable space, and let $\{A_{i}\}_{i\in I}$ be a discrete family of closed subsets of $X$. Then for every metric $d\in \mathrm{Met}(X)$, and for every family $\{e_{i}\}_{i\in I}$ of metrics with $e_{i}\in \mathrm{Met}(A_{i})$, there exists a metric $m\in \mathrm{Met}(X)$ satisfying t Moreover, if $X$ is completely metrizable, and if each $e_{i}\in \mathrm{Met}(A_{i})$ is a complete

Theorems & Definitions (67)

  • Theorem 1.1
  • Remark 1.1
  • Definition 1.1
  • Definition 1.2
  • Example 1.1
  • Definition 1.3
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Theorem 2.1
  • ...and 57 more