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Axioms for an n-metric Structure

Published online by Cambridge University Press:  20 November 2018

Kerry E. Grant
Kerry E. Grant*
Affiliation:
Southern Connecticut State College, New Haven, Connecticut
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From Euclid to Hilbert, and beyond, the primitive terms of geometry have been taken as “point,” “line,” etc., while “distance” plays a secondary role. The reversal of this situation is a modern development. Frechet [4], in 1906 first considered the properties of distance which should be formalized. The most significant contributions to the geometric properties of metric spaces have been by Menger [10] and Blumenthal [2; 3].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 1 , February 1973 , pp. 24 - 30
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Birkhoff, G., Metric foundations of geometry, Trans. Amer. Math. Soc. 55 (1944), 465.Google Scholar
2. Blumenthal, L. M., Distance Geometries, University of Missouri Studies 13, No. 2 (1938), 1–142.Google Scholar
3. Blumenthal, L. M., Theory and applications of distance geometry (Clarendon Press, Oxford, 1953).Google Scholar
4. Fréchet, M., Sur quelques points du calculfunctionnel, Rend. Cire. Mat. Palermo 22 (1906), 1–74.Google Scholar
5. Froda, A., Espaces p-métriaueetleurtopologie, C. R. Acad. Sci. Paris Sér. A-B 247 (1958), 849–52.Google Scholar
6. Freese, R. W. and Andalafte, E. Z., A characterization of 2-betweenness in 2-metric spaces, Can. J. Math. 18 (1966), 963–968.Google Scholar
7. Freese, R. W. and Andalafte, E. Z., Existence of 2-segments in 2-metric spaces, Fund. Math. 60 (1967), 201–208.Google Scholar
8. Gähler, S., 2-Metrische Raume und IhreTopologischeStruktur, Math. Nachr. 25 (1963), 115–148.Google Scholar
9. Grant, K. E., A 2-metric lattice structure, Doctoral Dissertation, St. Louis University, 1968.Google Scholar
10. Menger, K., UntersuchungeniiberAllgemeineMetrik, Math. Ann. 100 (1928), 75–163.Google Scholar
11. Murphy, G. P., Convexity and embedding in a class of 2-metrics, Doctoral Dissertation, St. Louis University, 1966.Google Scholar