Metric Segments in Gromov–Hausdorff class
https://doi.org/10.22405/2226-8383-2022-23-3-5-18
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 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.
Keywords
Gromov–Hausdorff distance,
class of all metric spaces,
von Neumann–Bernays– G¨odel axioms,
metric segment,
compact set.
Supporting agencies: The study was performed under the support of the Russian Foundation for Basic Research (project №19-01- 00775а) and the scholarship of the Theoretical Physics and Mathematics Advancement Foundation «BASIS» (grant №20-8-2-8-1).
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Review
For citations:
Borisova O.B. Metric Segments in Gromov–Hausdorff class. Chebyshevskii Sbornik. 2022;23(3):5-18. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-3-5-18
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