Menger and Nöbeling on pointless topology

Mathieu Bélanger, Jean-Pierre Marquis

DOI: http://dx.doi.org/10.12775/LLP.2013.009

Abstract


This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nöbeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nöbeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Menger's geometrical perspective was superseded by an algebraic one, a lattice-theoretical one to be precise. Second, Menger's bottom–up approach was replaced by a top–down one.


Keywords


pointless topology, Karl Menger, Georg Nöbeling, lattice theory, abstraction, generalization

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References


Albert, A.A., L.M. Graves, T. Fort, and P.A. Smith (eds.), Bulletin of the American Mathematical Society, vol. 46, 1940.

Arens, R., “Review of Grundlagen der analytischen Topologie”, Bulletin of the American Mathematical Society, 61 (1955), 6: 594–595.

Bénabou, J., “Treillis locaux et paratopologies”, in Séminaire Ehresmann. Topologie et de géométrie différentielle, vol. 1, 1957–1958, pp. 1–27.

Carathéodory, C., “Entwurf für eine Algebraisierung des Integralbegriffs”, in Gesammelte Mathematische Schriften, vol. IV, C.H. Beck’s Verlagsbuchhandlung, München, 1938, pp. 302–342.

Carathéodory, C., Algebraic Theory of Measure and Integration, Chelsea Publishing Company, New York, 1963.

Fréchet, M., “Sur quelques points du calcul fonctionel”, Rendiconti del Circolo Matematico di Palermo, 22 (1906): 1–74.

Geyer, W.-D., “Georg Nöbeling zum 100. Geburstag”, Akademie Aktuell. Zeitschrift der Bayerischen Akademie der Wissenschaften, 4 (2007), 23: 36–38.

Hausdorff, F., Grundzüge der Mengenlehre, Chelsea Publishing Company, New York, 1949.

Johnson, D. M., “The problem of the invariance of dimension in the growth of modern topology I”, Archive for History of Exact Sciences, 20 (1979), 2: 97–188.

Johnson, D. M., “The problem of the invariance of dimension in the growth of modern topology II”, Archive for History of Exact Sciences, 25 (1981), 2–3: 85–267.

Johnstone, P.T., “The point of pointless topology”, Bulletin of the American Mathematical Society, 8 (1983), 1: 41–53.

Johnstone, P.T., “Elements of the history of locale theory”, in C.E. Aull, and R. Lowen (eds.), Handbook of the History of General Topology, vol. 3, Kluwer Academic Publishers, Dordrecht, 2001, pp. 835–851.

Kass, S., “Karl Menger”, Notices of the American Mathematical Society, 43 (1996), 5: 558–561.

Kuratowski, K., “Sur l’opération A de l’Analysis Situs”, Fundamenta Mathematicae, 3 (1922): 182–199.

Menger, K., “Bemerkungen zu Grundlagenfragen IV. Axiomatik der endlichen Mengen und der elementargeometrischen Verknüpfungebeziehungen”, in L. Bieberbach, O. Blumenthal, and G. Faber

(eds.), Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 37, B.G. Teubner, Leipzig, 1928, pp. 309–324.

Menger, K., Dimensionstheorie, B.G. Teubner, Leipzig and Berlin, 1928.

Menger, K., “Bericht über metrische Geometrie”, in L. Bieberbach, O. Blumenthal, and G. Faber (eds.), Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 40, B.G. Teubner, Leipzig and Berlin, 1931, pp. 201–219.

Menger, K., “Analysis and metric geometry”, The Rice Institute Pamphlet, XXVII (1940), 1: 1–40.

Menger, K., “On algebra of geometry and recent progress in non-Euclidean geometry”, The Rice Institute Pamphlet, XXVII (1940), 1: 41–79.

Menger, K., “Topology without points”, The Rice Institute Pamphlet, XXVII (1940), 1: 80–107.

Menger, K., Géométrie générale, no. 124 in Mémorial des sciences mathématiques, Gauthier-Villars, Paris, 1954.

Milgram, A.N., “Partially ordered sets and topology”, Proceedings of the National Academy of Science of the United States of America, 26 (1940): 291–293.

Moore, R.L., “A set of axioms for plane analysis situs”, Fundamenta Mathematicae, 25 (1935), 1: 13–28.

Neumann, J. von, “Continuous geometry”, Proceedings of the National Academy of Science of the United States of America, 22 (1936), 2: 92–100.

Nöbeling, G., “Topologie der Vereine und Verbände”, Archiv der Mathematik, 1 (1948), 2: 154–159.

Nöbeling, G., “Limitentheorie in topologischen Vereinen und Verbände”, Journal für die reine und angewandte Mathematik, 191 (1953): 125–134.

Nöbeling, G., Grundlagen der analytischen Topologie, no. LXXII in Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer-Verlag, Berlin, 1954.

Stone, A. H., “Review of Grundlagen der analytischen Topologie”, Mathematical Reviews, (1955). MR0068197.

Stone, M.H., “The theory of representations for Boolean algebras”, Transactions of the American Mathematical Society, 40 (1936): 37–111.

Stone, M.H., “Applications of the theory of Boolean rings to general topology”, Transactions of the American Mathematical Society, 41 (1937), 3: 375–481.

Wald, A., “Über den allgemeinen Raumbegriff”, Ergebnisse eines mathematisches Kolloquium, 3 (1932): 6–11.








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