Fundamentals of theory of continuous Gromov-Hausdorff distance
https://doi.org/10.22405/2226-8383-2026-27-1-19-50
Abstract
The Gromov–Hausdorff distance (hereinafter referred to as the GH-distance) is a measure of non-isometricity of metric spaces. In this paper, we study a modification of this distance that also takes topological differences into account. The resulting function of pairs of metric spaces is called the continuous GH-distance. We show that many basic properties of the classical GH-distance also hold in the continuous case. However, the continuous GH-distance, distinguishing between topologies, can differ significantly from the classical one. We will provide numerous examples of this distinction and demonstrate the role of topological dimension here.
In particular, we will prove that the continuous GH-distance, like the classical one, is intrinsic, but, unlike the classical one, it is incomplete. Since we are dealing with all metric spaces, we will show, within the framework of the von Neumann-Bernays-G¨odel theory, how topological concepts can be transferred to proper classes.
About the Authors
Semeon Antonovich Bogatyi
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Alexey Avgustinovich Tuzhilin
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
1. Rieffel, M. A. 2003, “Gromov-Hausdorff Distance for Quantum Metric Spaces”, ArXiv e-prints, arXiv:math/0011063[math.OA].
2. Lim, S., Memoli, F. & Smith, Z. 2023, “The Gromov–Hausdorff distance between spheres”, Geometry & Topology, vol. 27, no. 9, pp. 3733–3800.
3. Lee, J. & Morales, C. A. 2022, “Gromov-Hausdorff Stability of Dynamical Systems and Applications to PDEs”, Birkh¨auser/Springer.
4. Mendelson, E. 1984, “Introduction to Mathematical Logic”, M.: Nauka.
5. Banach, T. 2023, “Classical set theory: theory of sets and classes”, ArXiv e-prints, arXiv:2006.01613v4[math.LO].
6. Gromov, M. 1981, “Structures m´etriques pour les vari´et´es riemanniennes”, Edited by Lafontaine and Pierre Pansu.
7. Gromov, M. 1999, “Metric structures for Riemannian and non-Riemannian spaces”, Birkh¨auser.
8. Burago, D., Burago, Yu. & Ivanov, S. 2001, “A Course in Metric Geometry”, Providence.
9. Bing, R. H. 1948, “A homogeneous indecomposable plane continuum”, Duke Math. J., vol. 15, pp. 729–742.
10. Bogatyy, S. A. & Tuzhilin, A. A. 2021, “Gromov–Hausdorff class: its completeness and cloud geometry”, ArXiv e-prints, arXiv:2110.06101[math.MG].
11. Bogatyy, S. A. & Tuzhilin, A. A. 2023, “Action of similarity transformation on families of metric spaces”, Itogi Nauki i Tekhn., vol. 223, pp. 3–13.
12. Bogataya, S. I., Bogatyy, S. A., Redkozubov, V. V. & Tuzhilin, A. A. 2023, “Clouds in Gromov– Hausdorff Class: their completeness and centers”, Topology and its Applications, vol. 329.
13. Bogatyy, S. A. & Tuzhilin, A. A. 2021, “Continuous Gromov–Hausdorff class: its completeness and cloud geometry”, ArXiv e-prints, arXiv:2110.06101[math.MG].
14. Fet, A. I. 1954, “A generalization of the Lyusternik-Shnirelman theorem on coverings of spheres and some related theorems”, DAN, vol. 95, no. 6, pp. 1149–1151.
15. Vikhrov, A. A. 2025, “The problem of constructing geodesics in the Gromov–Hausdorff class: an optimal Hausdorff implementation does not always exist”, Chebyshevski sb., vol. 26, no. 2, pp. 49–60.
16. Vikhrov, A. 2025, “Geometry of linear and nonlinear geodesics in the proper Gromov–Hausdorff class”, Matematicki vesnik, pp. 1–17.
17. Stone, A. H. 1948, “Paracompactness and product spaces”, Bull. Amer. Math. Soc., vol. 54, pp. 977–982.
18. Engelking, R. 1985, “General Topology”, Warszawa.
19. Munkres, J. R., 2000, “Topology”, 2nd Edition, Prentice Hall, Upper Saddle River.
20. Dowker, C. H. 1947, “Mapping theorems for non-compact spaces”, Amer. J. Math., vol. 69, pp.
21. –242.
22. Nagami, K. & Roberts, J. H. 1965, “Metric-dependent dimension functions”, Proc. Amer. Math. Soc., vol. 16, no. 4, pp. 601–604.
23. Illanes, A. & Nadler, S. Jr. 1999, “Hyperspaces”, Marcel Dekker, New York.
24. Knaster, B. 1922, “Un continu dont tout sous-continu est indecomposable”, Fund. Math., vol. 3, pp. 247–286.
25. Moise, E. E. 1948, “An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua”, Trans. Amer. Math. Soc., vol. 63, pp. 581–594.
26. Bing, R. H. 1951, “Concerning hereditarily indecomposable continua”, Pacific J. Math., vol. 1, pp. 43–51.
27. Cook, H. 1967, “Continua which admit only the identity mapping onto non-degenerate subcontinua”, Fundamenta Mathematicae, vol. 60, pp. 241–249.
28. Borzov, S. I., Ivanov, A. O. & Tuzhilin, A. A. 2020, “Extendability of Metric Segments in Gromov-Hausdorff Distance”, ArXiv e-prints, arXiv:2009.00458[math.MG].
Review
For citations:
Bogatyi S.A., Tuzhilin A.A. Fundamentals of theory of continuous Gromov-Hausdorff distance. Chebyshevskii Sbornik. 2026;27(1):19-50. (In Russ.) https://doi.org/10.22405/2226-8383-2026-27-1-19-50
Views:
409
JATS XML
Email this article 