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Research Article
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The Unique Midpoint Property in Ultrametric Spaces

Year 2025, Volume: 9 Issue: 2, 187 - 190

Mehmet Vural

Abstract

In this paper, we investigate ultrametric spaces that satisfy the Unique Midpoint Property (UMP), a condition where every pair of distinct points has a unique midpoint equidistant from both. By analyzing the interaction between the strong triangle inequality of ultrametric spaces and the requirements of the UMP, we derive fundamental constraints on the structure of such spaces. In particular, we prove that in any ultrametric space with the UMP, the distance between any two distinct points equals the distance from each to their unique midpoint. Our main result shows that any ultrametric space satisfying the UMP must be trivial in structure it can only consist of either a single point or exactly three points.

Keywords

Metric Ultrametric Unique Midpoint Property Equidistance Midset

References

  • [1] Lemin, A. (2003). On ultrametrization of general metric spaces. Proceedings of the American Mathematical Society, 131(3): 979-989.
  • [2] Kitai, Y. (2005). Unique midpoint property in metrizable spaces: a survey. Memoirs of the Faculty of Science and Engineering Shimane University Series B Mathematical Science, 38: 31-38 . [3] Hattori, Y. and Ohta, H. (1993). A metric characterization of a subspace of the real line. Topology Proceedings, 18: 75-87.
  • [4] Nadler Jr., S.B. (1976). An embedding theorem for certain spaces with an equidistant property. Proceedings of the American Mathematical Society, 59(1): 179-183.
  • [5] Priestley, H.A. (1970). Semilattices and complete convexity. Bulletin of the London Mathematical Society, 2(2): 177–182.
There are 4 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Research Articles
Authors

Mehmet Vural 0000-0002-0977-7479

Early Pub Date August 20, 2025
Publication Date August 30, 2025
Submission Date February 20, 2025
Acceptance Date April 18, 2025
Published in Issue Year 2025 Volume: 9 Issue: 2

Cite

APA Vural, M. (2025). The Unique Midpoint Property in Ultrametric Spaces. Journal of Innovative Science and Engineering, 9(2), 187-190. https://doi.org/10.38088/jise.1644123
AMA Vural M. The Unique Midpoint Property in Ultrametric Spaces. JISE. August 2025;9(2):187-190. doi:10.38088/jise.1644123
Chicago Vural, Mehmet. “The Unique Midpoint Property in Ultrametric Spaces”. Journal of Innovative Science and Engineering 9, no. 2 (August 2025): 187-90. https://doi.org/10.38088/jise.1644123.
EndNote Vural M (August 1, 2025) The Unique Midpoint Property in Ultrametric Spaces. Journal of Innovative Science and Engineering 9 2 187–190.
IEEE M. Vural, “The Unique Midpoint Property in Ultrametric Spaces”, JISE, vol. 9, no. 2, pp. 187–190, 2025, doi: 10.38088/jise.1644123.
ISNAD Vural, Mehmet. “The Unique Midpoint Property in Ultrametric Spaces”. Journal of Innovative Science and Engineering 9/2 (August2025), 187-190. https://doi.org/10.38088/jise.1644123.
JAMA Vural M. The Unique Midpoint Property in Ultrametric Spaces. JISE. 2025;9:187–190.
MLA Vural, Mehmet. “The Unique Midpoint Property in Ultrametric Spaces”. Journal of Innovative Science and Engineering, vol. 9, no. 2, 2025, pp. 187-90, doi:10.38088/jise.1644123.
Vancouver Vural M. The Unique Midpoint Property in Ultrametric Spaces. JISE. 2025;9(2):187-90.
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