Commentary on Menger’s Work on Algebra (original) (raw)

Selecta Mathematica, 2003

Abstract

When Menger’s book on curve theory [23], written in co-operation with Georg Nobeling, appeared in 1932 he already had over 60 publications to his credit. Among them were his group theoretical investigations, an offshoot of his seminal paper “Untersuchungen uber allgemeine Metrik” [18]. The Mathematisches Kolloquium at the University of Vienna discussed the ideas developed in this paper and in this way other young mathematicians, especially Abraham Wald (cf. [36]), Olga Taussky, Franz Alt and Gustav Beer, became actively interested in Menger’s distance geometry. At the 13th meeting of the Kolloquium held on 14 March 1930, Menger presented new results under the heading “u ber eine metrische Geometrie in Gruppen” [19]. They were published in the Mathematische Zeitschrift as “Beitrage zur Gruppentheorie. I. uber eine Gruppenmetrik”, [21]1. The second of Menger’s three investigations in [18], “Die euklidische Metrik”, opens with a proof of the following theorem discovered by M. M. Biedermann: a connected metric space M is homeomorphic with a subspace of l i if, for each triple (a, b, c) of distinct points of M, one of the points lies between the other two. 2 However, Menger observed that Biedermann’s condition does not characterise metric spaces which are isometric (“abstandsgleich” in [18], “kongruent” in [20]) to some subspace of ℝ1. Thus he posed the problem of finding necessary and sufficient conditions for a semimetric space3 to be isometrically embeddable (“abstandstreu einbettbar”) into — or, more specifically, to be isometric to —the euclidean n-space ℝ n .

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