Metric Segments in Gromov--Hausdorff class

Olga Borisova

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arXiv · 2009.13273

Metric Segments in Gromov--Hausdorff class

Olga Borisova

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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--Gödel (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.

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BibTeX RIS

Olga Borisova. 2020-09-28. Metric Segments in Gromov--Hausdorff class. https://arxiv.org/abs/2009.13273

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