Metric Segments in Gromov–Hausdorff class

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–Gödel (NBG) axiomatic set theory, a proper class is a “monster collection”, e.g., the collection of all 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. If the distance between the segment endpoints is zero, then the metric segment is a set. In addition, we show that the restriction of a non-degenerated metric segment to compact metric spaces is a non-compact set.