Metric Segments in Gromov--Hausdorff class

TL;DR

This paper investigates the structure of metric segments within the Gromov-Hausdorff class of all metric spaces, revealing their size and topological properties, especially in the context of proper classes and compact spaces.

Contribution

It proves that metric segments in the class of all metric spaces are proper classes if they contain a space at positive distance from endpoints, and shows restrictions on compact spaces.

Findings

Metric segments in the proper class of all metric spaces are proper classes.

Restriction of non-degenerate segments to compact spaces is non-compact.

Provides set-theoretic insights into the structure of metric segments.

Abstract

We study properties of metric segments in the class of all metric spaces considered up to an isometry, endowed with Gromov--Hausdorff distance. On the isometry classes of all compact metric spaces, the Gromov-Hausdorff distance is a metric. A metric segment is a class that consists of points lying between two given ones. By von Neumann--Bernays--G\"odel (NBG) axiomatic set theory, a proper class is a "monster collection", e.g., the collection of all cardinal sets. We prove that any metric segment in the proper class of isometry classes of all metric spaces with the Gromov-Hausdorff distance is a proper class if the segment contains at least one metric space at positive distances from the segment endpoints. In addition, we show that the restriction of a non-degenerated metric segment to compact metric spaces is a non-compact set.