The Unique Midpoint Property in Ultrametric Spaces

Mehmet Vural

doi:10.38088/jise.1644123

The Unique Midpoint Property in Ultrametric Spaces

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

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.

Details

Primary Language

English

Subjects

Applied Mathematics (Other)

Journal Section

Research Article

Authors

Mehmet Vural ^*
0000-0002-0977-7479
Türkiye

Early Pub Date

August 20, 2025

Publication Date

December 15, 2025

Submission Date

February 20, 2025

Acceptance Date

April 18, 2025

Published in Issue

Year 2025 Volume: 9 Number: 2

DOI

https://doi.org/10.38088/jise.1644123

IZ

https://izlik.org/JA99TN39MR

Cite

RIS / Bibtex

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

1.Vural M. The Unique Midpoint Property in Ultrametric Spaces. JISE. 2025;9(2):187-190. doi:10.38088/jise.1644123

Chicago

Vural, Mehmet. 2025. “The Unique Midpoint Property in Ultrametric Spaces”. Journal of Innovative Science and Engineering 9 (2): 187-90. https://doi.org/10.38088/jise.1644123.

EndNote

Vural M (December 1, 2025) The Unique Midpoint Property in Ultrametric Spaces. Journal of Innovative Science and Engineering 9 2 187–190.

IEEE

[1]M. Vural, “The Unique Midpoint Property in Ultrametric Spaces”, JISE, vol. 9, no. 2, pp. 187–190, Dec. 2025, doi: 10.38088/jise.1644123.

ISNAD

Vural, Mehmet. “The Unique Midpoint Property in Ultrametric Spaces”. Journal of Innovative Science and Engineering 9/2 (December 1, 2025): 187-190. https://doi.org/10.38088/jise.1644123.

JAMA

1.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, Dec. 2025, pp. 187-90, doi:10.38088/jise.1644123.

Vancouver

1.Mehmet Vural. The Unique Midpoint Property in Ultrametric Spaces. JISE. 2025 Dec. 1;9(2):187-90. doi:10.38088/jise.1644123