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A factorization of metric spaces

Yoshito Ishiki

TL;DR

The paper establishes a factorization of a metrizable space $X$ by embedding it into a product $F\times\Lambda$, where $F$ is a closed subset with $X\setminus F$ zero-dimensional and $\Lambda$ is a metrizable $0$-dimensional space, via an Engelking-type retraction. This embedding enables the construction of metric and ultrametric extensors $\Xi$ and $\Sigma$ that extend metrics from $F$ to $X$ while preserving key properties such as completeness, properness, ultrametricity, fractal dimensions, and large-scale structure. It unifies previous extension theorems for ultrametrics and proper ultrametrics by exploiting the product-structure and the metric quotient $X_{/F}$. The framework also provides a systematic way to realize prescribed fractal dimensions through $\mathrm{FD}(X;\mathbf{a})$ and $\mathrm{TD}(X;\mathbf{b})$, and derives product-dimension inequalities that facilitate analysis of metric products.

Abstract

We first prove that for every metrizable space $X$, for every closed subset $F$ whose complement is zero-dimensional, the space $X$ can be embedded into a product space of the closed subset $F$ and a metrizable zero-dimensional space as a closed subset. Using this theorem, we next show the existence of extensors of metrics and ultrametrics, which preserve properties of metrics such as the completeness, the properness, being an ultrametrics, its fractal dimensions, and large scale structures. This result contains some of the author's extension theorems of ultrametrics.

A factorization of metric spaces

TL;DR

The paper establishes a factorization of a metrizable space by embedding it into a product , where is a closed subset with zero-dimensional and is a metrizable -dimensional space, via an Engelking-type retraction. This embedding enables the construction of metric and ultrametric extensors and that extend metrics from to while preserving key properties such as completeness, properness, ultrametricity, fractal dimensions, and large-scale structure. It unifies previous extension theorems for ultrametrics and proper ultrametrics by exploiting the product-structure and the metric quotient . The framework also provides a systematic way to realize prescribed fractal dimensions through and , and derives product-dimension inequalities that facilitate analysis of metric products.

Abstract

We first prove that for every metrizable space , for every closed subset whose complement is zero-dimensional, the space can be embedded into a product space of the closed subset and a metrizable zero-dimensional space as a closed subset. Using this theorem, we next show the existence of extensors of metrics and ultrametrics, which preserve properties of metrics such as the completeness, the properness, being an ultrametrics, its fractal dimensions, and large scale structures. This result contains some of the author's extension theorems of ultrametrics.
Paper Structure (14 sections, 20 theorems, 15 equations)

This paper contains 14 sections, 20 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities
  4. Topological dimension
  5. Metric quotient spaces
  6. Retractions
  7. Embedding into a product space
  8. Applications
  9. Fractal dimensions
  10. The Hausdorff dimension
  11. The packing dimension
  12. Box dimensions
  13. The Assouad dimension
  14. Extensors

Key Result

Lemma 2.1

Let $(X, d)$ and $(Y, e)$ be metric spaces. If both $d$ and $e$ are proper, then so are $d\times_{1} e$ and $d\times_{\infty} e$.

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Remark 2.1
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 35 more