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An Embedding, An Extension, and An Interpolation of Ultrametrics\(^*\)

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Abstract

The notion of ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the Arens-Eells isometric embedding theorem of metric spaces, the Hausdorff extension theorem of metrics, the Niemytzki-Tychonoff characterization theorem of the compactness, and the author’s interpolation theorem of metrics and theorems on dense subsets of spaces of metrics.

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References

  1. R. F. Arens and J. Eells, Jr, “On embedding uniform and topological spaces,” Pacific. J. Math. 6, 397–403 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  2. S. A. Bogatyi, “Metrically homogeneous spaces,” Russian Math. Surveys 57 (2), 221–240 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  3. N. Brodskiy, J. Dydak, J. Higes and A. Mitra, “Dimension zero at all scales,” Topology Appl. 154, 2729–2740 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics 33 (Amer. Math. Soc., Providence, RI, 2001).

    MATH  Google Scholar 

  5. L. W. Cohen and C. Goffman, “On the metrization of uniform space,” Proc. Amer. Math. Soc. 1 (6), 750–753 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Conant, “Distance structures for generalized metric spaces,” Ann. Pure Appl. Logic 168 (3), 622–650 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. B. Comicheo and K. Shamseddine, “Summary on non-Archimedean valued field,” in Advances in Ultrametric Analysis, Contemporary Mathematics 704, 1–36 (Amer. Math. Soc., Providence, RI, 2018).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. B. Comicheo, “Generalized open mapping theorem for \(X\)-normed spaces,” \(p\)-Adic Numbers Ultrametric Anal. Appl. 11 (2), 135–150 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Corazza, “Introduction to metric-preserving functions,” Amer. Math. Monthly 106 (4), 309–323 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Dancis, “Each closed subset of metric space \(X\) with \(\mathrm{Ind}\ X=0\) is a retract,” Houston J. Math. 19 (4), 541–550 (1993).

    MathSciNet  MATH  Google Scholar 

  11. C. Delhommé, C. Laflamme, M. Pouzet and N. Sauer, “Indivisible ultrametric space,” Topology Appl. 155 (14), 1462–1478 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Dordovskyi, O. Dovgoshey and E. Petrov, “Diameter and diametrical pairs of points in ultrametric spaces,” \(p\)-Adic Numbers Ultrametric Anal. Appl. 3 (4), 253–262 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. O. Dovgoshey, “On ultrametric-preserving functions,” Math. Slovaca 70, 173–182 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  14. O. Dovgoshey, O. Martio and M. Vuorinen, “Metrization of weighted graphs,” Ann. Comb. 17, 455–476 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. A. Dovgoshey and E. A. Petrov, “Subdominant pseudoultrametric on graphs,” Sb. Math. 204, 1131–1151 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  16. O. Dovgoshey and V. Shcherbak, “The range of ultrametrics, compactness, and separability,” arXiv:2102.10901v2, (2102).

  17. R. Ellis, “Extending continuous functions on zero-dimensional spaces,” Math. Ann. 186, 114–122 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Engelking, General Topology (PWN, Warsawa, 1977).

    MATH  Google Scholar 

  19. K. Funano, “Embedding proper ultrametric spaces into \(\ell^p\) and its application to nonlinear Dvoretzky’s theorem,” arXiv:1203.1761, (2012).

  20. S. Gao and C. Shao, “Polish ultrametric Urysohn space and their isometry groups,” Topology Appl. 158 (3), 492–508 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  21. J. de Groot, “Non-Archimedean metrics in topology,” Proc. Amer. Math. Soc. 7, 948–953 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  22. J. de Groot, “Some special metrics in general topology,” Colloq. Math. 6, 283–286 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Gromov, with appendices by M. Katz, P. Pansu and S. Semmes, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Math. 152 (Birkhauser, 1999).

    Google Scholar 

  24. F. Hausdorff, “Erweiterung einer Homöorphie,” Fund. Math. 16, 353–360 (1930).

    Article  MATH  Google Scholar 

  25. F. Hausdorff, “Erweiterung einer stetigen Abbildung,” Fund. Math. 30, 40–47 (1938).

    Article  MATH  Google Scholar 

  26. J. Heinonen, Lectures on Analysis on Metric Spaces (Springer-Verlag, New York, 2001).

    Book  MATH  Google Scholar 

  27. Y. Ishiki, “Quasi-symmetric invariant properties of Cantor metric spaces,” Ann. Inst. Fourier (Grenoble) 69 (6), 2681–2721 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Ishiki, “On the Assouad dimension and convergence of metric spaces,” arXiv:1911.07455, (2019).

  29. Y. Ishiki, “An interpolation of metrics and spaces of metrics,” arXiv:2003.13277, (2020).

  30. C. Kuratowski, “Remarques sur les transformations continues des espaces métriques,” Fund. Math. 30, 48–49 (1938).

    Article  MATH  Google Scholar 

  31. A. J. Lemin and V. A. Lemin, “On a universal ultrametric space,” Topology Appl. 103, 339–345 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  32. A. J. Lemin, “Isometric embedding ultrametric (non-Archimedean) spaces in Hilbert space and Lebesgue space,” in \(p\)-Adic Functional Analysis, Lect. Notes Pure Appl. Math. 222, 203–218 (Dekker, New York, 2001).

    MathSciNet  MATH  Google Scholar 

  33. J. M. Mackay and J. T. Tyson, Conformal Dimension Theory: Theory and Application, Univ. Lecture Ser. 54 (Amer. Math. Soc., 2010).

    MATH  Google Scholar 

  34. M. Megrelishvili and M. Shlossberg, “Free non-Archimedean topological groups,” Comment. Math. Univ. Carol. 54 (2), 273–312 (2013).

    MathSciNet  MATH  Google Scholar 

  35. E. Michael, “Continuous selections. I,” Ann. Math. (2) 63 (2), 361–382 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Michael, “Continuous selections. II,” Ann. Math. (2) 64 (3), 562–580 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  37. E. Michael, “Selected selection theorems,” Amer. Math. Monthly 63 (4), 233–238 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  38. V. Niemytzki and A. Tychonoff, “Beweis des Satzes, dass ein metrisierbarer Raum dann und nur dann kompakt ist, wenn er in jeder Metrik vollständig ist,” Fund. Math. 12, 118–120 (1928).

    Article  MATH  Google Scholar 

  39. H. Ochsenius and W. H. Schikhof, “Banach spaces over fields with an infinite rank valuation,” in \(p\)-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math. 207, 233–293 (Marcel Dekker, 1999).

    MathSciNet  MATH  Google Scholar 

  40. D. Qiu, “Geometry of Non-Archimedean Gromov-Hausdorff distance,” \(p\)-Adic Numbers Ultrametric Anal. Appl. 1, 317–337 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  41. A. M. Robert, A Course in \(p\)-adic Analysis, Graduate Text in Math. 198 (Springer-Verlag, 2000).

    Book  MATH  Google Scholar 

  42. A. R. Pears, Dimension Theory of General Spaces (Cambridge Univ. Press, 1975).

    MATH  Google Scholar 

  43. P. Pongsriiam and I. Termwuttipong, “Remarks on ultrametrics and metric-preserving functions,” Abstr. Appl. Anal. 2014, 1–9 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  44. W. H. Schikhof, “Isometrical embeddings of ultrametric spaces into non-Archimedean valued fields,” Indag. Math. 46, 51–53 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  45. R. Sikorski, “Remarks on some topological spaces of high power,” Fund. Math. 33, 125–136 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  46. I. S. Stares and J. E. Vaughan, “The Dugundji extension property can fail in \(\omega_{\mu}\)-metrizable spaces,” Fund. Math. 150, 11–16 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  47. I. Z. Stasyuk, “On a homogeneous operator extending partial ultrametrics,” Matematychni Studii 22 (1), 73–78 (2004).

    MathSciNet  MATH  Google Scholar 

  48. I. Z. Stasyuk and E. D. Tymchatyn, “Extending pairs of metrics,” Matematychni Studii 35 (2), 215–224 (2011).

    MathSciNet  MATH  Google Scholar 

  49. A. H. Stone, “Paracompactness and product spaces,” Bull. Amer. Math. Soc. 54 (10), 977–982 (1948).

    Article  MathSciNet  MATH  Google Scholar 

  50. A. F. Timan and I. A. Vestfrid, “A universality property of Hilbert spaces,” Soviet Math. Dokl. 20, 485–486 (1979).

    MATH  Google Scholar 

  51. A. F. Timan and I. A. Vestfrid, “Any separable ultrametric space is isometrically embeddable in \(\ell^2\),” Funct. Anal. Appl. 17, 70–71 (1983).

    Article  MathSciNet  Google Scholar 

  52. H. Toruńczyk, “A simple proof of Hausdorff’s theorem on extending metrics,” Fund. Math. 77, 191–193 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  53. E. D. Tymchatyn and M. M. Zarichnyi, “A note on operators extending partial ultrametrics,” Comment. Math. Univ. Carol. 46 (3), 515–524 (2005).

    MathSciNet  MATH  Google Scholar 

  54. J. E. Vaughan, “Universal ultrametric spaces of smallest weight,” Topology Proc. 24, 611–619 (1999).

    MathSciNet  MATH  Google Scholar 

  55. I. A. Vestfrid, “On the universal spaces,” Ukrainian Math. J. 46 (12), 1890–1898 (1994).

    Article  MathSciNet  Google Scholar 

  56. Z. Wan, “A novel construction of Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametric,” arXiv:2007.08105, (2020).

  57. S. Warner, Topological Fields, North Holland, Mathematics Studies 157 (North-Holland-Amsterdam, London, New York, Tokyo, 1993).

    MATH  Google Scholar 

  58. S. Willard, General Topology (Dover Publications, 2004).

    MATH  Google Scholar 

  59. I. Zarichnyi, “Gromov-Hausdorff ultrametric,” arXiv:math/0511436v1, (2005).

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Acknowledgments

The author would like to thank Professor Koichi Nagano for his advice and constant encouragement. The author would also like to thank the referee for helpful comments and suggestions.

Funding

The author was supported by JSPS KAKENHI Grant Number 18J21300.

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Authors and Affiliations

  1. Graduate School of Pure and Applied Sciences, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan

    Yoshito Ishiki

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Ishiki, Y. An Embedding, An Extension, and An Interpolation of Ultrametrics\(^*\). P-Adic Num Ultrametr Anal Appl 13, 117–147 (2021). https://doi.org/10.1134/S2070046621020023

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  • DOI: https://doi.org/10.1134/S2070046621020023

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