Metric Segments in Gromov--Hausdorff class

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.