We give an elementary proof of the classical Menger result according to which any metric space X that consists of more than four points is isometrically imbedded into \( \mathbb{R} \) if every three-point subspace of X is isometrically imbedded into \( \mathbb{R} \). A series of corollaries of this theorem is obtained. We establish new criteria for finite metric spaces to be isometrically imbedded into \( \mathbb{R} \).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 10, pp. 1319 – 1328, October, 2009.
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Dovgoshei, A.A., Dordovskii, D.V. Betweenness relation and isometric imbeddings of metric spaces. Ukr Math J 61, 1556–1567 (2009). https://doi.org/10.1007/s11253-010-0297-7
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DOI: https://doi.org/10.1007/s11253-010-0297-7