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A Characterization of 2-Betweenness in 2-Metric Spaces

Published online by Cambridge University Press:  20 November 2018

Raymond W. Freese  and
Edward Z. Andalafte
Raymond W. Freese
Affiliation:
St. Louis University and University of Missouriat St. Louis
Edward Z. Andalafte
Affiliation:
St. Louis University and University of Missouriat St. Louis
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The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 963 - 968
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Gähler, S., 2-metrische Räume und ihre topologische Struktur, Math. Nachr., 26 (1963), 115–148.Google Scholar
2. Froda, A., Espaces p-métriques et leur topologie, C. R. Acad. Sci. Paris, 247 (1958), 849–852.Google Scholar
3. Menger, K., Untersuchungen über allgemeine Metrik, Math. Ann., 100 (1928), 75–163.Google Scholar
4. Blumenthal, L. M., Theory and applications of distance geometry (Oxford, 1953).Google Scholar
5. Wald, A., Axiomatik des Zwischenbegriffes in metrischen Räumen, Math. Ann., 104 (1931), 476–484.Google Scholar