Chain development of metric compacts

TL;DR

This paper investigates the conditions under which a metric compact can be mapped into the real line while preserving chain distances, providing a criterion for existence and analyzing diameter properties of such mappings.

Contribution

It introduces a criterion for the existence of chain developments of metric compacts and characterizes when their diameters are preserved.

Findings

A criterion for the existence of chain developments in metric compacts.

The diameter of chain developments is preserved iff the compact is countable.

Provides insights into the structure of chain distances in metric spaces.

Abstract

Chain distance between points in a metric space is defined as the infimum of epsilon such that there is an epsilon-chain connecting these points. We call a mapping of a metric compact into the real line a chain development if it preserves chain distances. We give a criterium of existence of the chain development for metric compacts. We prove the diameter of any chain development of a given compact to be the same iff the compact is countable.