A factorization of metric spaces

TL;DR

The paper establishes a new factorization theorem for metrizable spaces and demonstrates the existence of metric and ultrametric extensions that preserve key properties, advancing the understanding of metric space embeddings.

Contribution

It introduces a novel factorization theorem for metrizable spaces and constructs metric and ultrametric extensions that maintain important properties.

Findings

Embedding of spaces into product spaces with zero-dimensional complements

Existence of metric and ultrametric extensions preserving properties

Extension theorems for ultrametrics with various invariants

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.