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The emergence of open sets, closed sets, and limit points in analysis and topology. (English) Zbl 1153.54001

This paper traces the history of the concepts of open set, closed set, and limit point of a set.
From the historical point of view, the primary notion was that of a limit point. Limit points were introduced by Weierstrass in his (unpublished) lectures on complex function theory, while the first publication dealing with limit points was due to Cantor. With the help of limit points, Cantor was able to define derived sets and closed sets, which were crucial in his studies of trigonometric series.
The story of open sets was more complicated. Peano distinguished between interior, exterior and boundary points of a given set. Instead of speaking about open sets, Jordan and Borel were also referring to sets whose every point is interior. Dedekind introduced the term Körper equivalent to an open set, but he didn’t make his ideas public. In his dissertation devoted to semicontinuous functions, Baire proposed a definition of an “open domain of \(n\) dimensions”. Lebesgue’s dissertation also contained definition of open sets, together with the proof that every open set is Borel-measurable.
A large part of the paper is devoted to the process of axiomatization of “abstract spaces”. The history starts with Fréchet’s L-spaces (based on the primitive notion of a limit of a sequence) and metric spaces, and proceeds to Hausdorff’s neigborhood spaces, Kuratowski’s closure spaces and derived spaces. Open sets, which are crucial in the modern definition of a topological space, were the primary notion for Tietze, Aleksandrov and Sierpiński. These authors were also considering the effects of adding various forms of separation axioms.
Abstract spaces became important in real analysis and combinatorial topology, but different authors proposed various axiomatizations. The modern definition of topological spaces survived as the fittest, due to the great influence of the textbooks written by the Bourbaki group, and of J. L. Kelley’s “General topology” (1955; Zbl 0066.16604). More general definitions proposed by Sierpiński, Aleksandrov and Hopf didn’t prove so fruitful and are almost forgotten today.
Apart from the history of the basic topological notions, the paper focuses on two important theorems, namely the Bolzano-Weierstrass theorem and the Heine-Borel theorem. The author also describes the attempts to define the notion of connected set and discusses various meanings assigned to the terms “Gebiet”, “continuum”, “domain” and “region” by different authors.

MSC:

54-03 History of general topology
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
57-03 History of manifolds and cell complexes

Citations:

Zbl 0066.16604
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References:

[1] Ahlfors, L. V., Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (1979), McGraw-Hill: McGraw-Hill New York · Zbl 0395.30001
[2] Aleksandrov, P., Zur Begründung der \(n\)-dimensionalen mengentheoretischen Topologie, Mathematische Annalen, 94, 296-308 (1925) · JFM 51.0451.02
[3] Aleksandrov, P. S.; Hopf, H., Topologie (1935), Springer-Verlag: Springer-Verlag Berlin
[4] (Aull, C. E.; Lowen, R., Handbook of the History of General Topology, 3 vols (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht) · Zbl 0980.00038
[5] Baire, R., Sur les fonctions de variables réelles, Annali di matematica pura ed applicata (3), 3, 1-123 (1899) · JFM 30.0359.01
[6] Beaulieu, L., 1989. Bourbaki: Une histoire du groupe de mathématiciens français et de ses travaux (1934-1944). Doctoral dissertation. Université de Montréal; Beaulieu, L., 1989. Bourbaki: Une histoire du groupe de mathématiciens français et de ses travaux (1934-1944). Doctoral dissertation. Université de Montréal
[7] Bolzano, B., Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (1817), Gottlieb Haase: Gottlieb Haase Prague · JFM 37.0446.05
[8] Bolzano, B., Paradoxes of the Infinite (1851/1950), Routledge and Kegan Paul: Routledge and Kegan Paul London, (D.A. Steele, Trans.)
[9] Borel, E., Sur quelques points de la théorie des fonctions, Annales scientifiques de l’Ecole Normale Supérieure (3), 12, 9-55 (1895) · JFM 26.0429.03
[10] Borel, E., Leçons sur la théorie des fonctions (1898), Gauthier-Villars: Gauthier-Villars Paris · JFM 29.0336.01
[11] Borel, E., Leçons sur les fonctions monogènes uniformes d’une variable complexe (1917), Gauthier-Villars: Gauthier-Villars Paris · JFM 46.0465.03
[12] Bourbaki, N., Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques, Actualités scientifiques et industrielles, vol. 858 (1940), Hermann: Hermann Paris · JFM 66.1357.01
[13] Bourbaki, N., Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques, Actualités scientifiques et industrielles, vol. 1142 (1951), Hermann: Hermann Paris · JFM 66.1357.01
[14] Cantor, G., Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, Mathematische Annalen, 5, 123-132 (1872), Reprinted in [1932, 92-102]. Pagination agrees with the reprint · JFM 04.0101.02
[15] Cantor, G., Über einen Satz aus der Theorie der stetigen Mannigfaltigkeiten, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physicalische Klasse, 127-135 (1879), Reprinted in [1932, 134-138]. Pagination agrees with the reprint · JFM 11.0352.01
[16] Cantor, G., Über unendliche lineare Punktmannichfaltigkeiten. 3, Mathematische Annalen, 20, 113-121 (1882), Reprinted in [1932, 149-157]. Pagination agrees with the reprint · JFM 14.0433.01
[17] Cantor, G., Über unendliche lineare Punktmannichfaltigkeiten. 4, Mathematische Annalen, 21, 51-58 (1883), Reprinted in [1932, 157-164]. Pagination agrees with the reprint
[18] Cantor, G., Über unendliche lineare Punktmannichfaltigkeiten. 5, Mathematische Annalen, 21, 545-586 (1883), Reprinted in [1932, 165-209]. Pagination agrees with the reprint
[19] Cantor, G., Über unendliche lineare Punktmannichfaltigkeiten. 6, Mathematische Annalen, 23, 453-488 (1884), Reprinted in [1932, 210-244]. Pagination agrees with the reprint · JFM 16.0459.01
[20] Cantor, G., Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (1932), Springer-Verlag: Springer-Verlag Berlin, (Ernst Zermelo, Ed.) · JFM 58.0043.01
[21] Carathéodory, C., Vorlesungen über reelle Funktionen (1918), Teubner: Teubner Leipzig · JFM 46.0376.12
[22] Cavaillès, J., Philosophie mathématique (1962), Hermann: Hermann Paris · Zbl 0105.24408
[23] Dauben, J. W., The trigonometric background to Georg Cantor’s theory of sets, Archive for History of Exact Sciences, 7, 181-216 (1970) · Zbl 0217.00404
[24] Dauben, J. W., Georg Cantor: His Mathematics and Philosophy of the Infinite (1979), Harvard University Press: Harvard University Press Cambridge, MA · Zbl 0463.01009
[25] de la Vallée Poussin, C., Intégrales de Lebesgue, fonctions d’ensemble, classes de Baire (1916), Gauthier-Villars: Gauthier-Villars Paris · JFM 46.1519.01
[26] Dedekind, R., Gesammelte mathematische Werke, vol. 2 (1931), Vieweg: Vieweg Braunschweig, (R. Fricke, E. Noether, Ö. Ore, Eds.) · JFM 57.0036.01
[27] Denjoy, A., Continu et discontinu, Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Paris, 151, 138-140 (1910) · JFM 41.0102.02
[28] Dini, U., Fondamenti per la teorica della funzioni di variabili reali (1878), Nistri: Nistri Pisa · JFM 10.0274.01
[29] Dugac, P., Richard Dedekind et les fondements des mathématiques (1976), Vrin: Vrin Paris · Zbl 0321.01027
[30] Epple, M., Zum Begriff des topologischen Raumes, (Brieskorn, E.; etal., Felix Hausdorff Gesammelte Werke, Band II: Grundzüge der Mengenlehre (2002), Springer-Verlag: Springer-Verlag Berlin), 675-744
[31] Ferreirós, J., Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (1999), Birkhäuser: Birkhäuser Basel/Boston/Berlin · Zbl 0934.03058
[32] Fréchet, M., Généralisation d’un théorème de Weierstrass, Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Paris, 139, 848-850 (1904) · JFM 35.0389.02
[33] Fréchet, M., Sur les fonctions limites et les opérations fontionnelles, Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Paris, 140, 27-29 (1905) · JFM 36.0449.02
[34] Fréchet, M., Sur quelques points du Calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, 22, 1-74 (1906) · JFM 37.0348.02
[35] Fréchet, M., Sur les ensembles abstraits, Annales scientifiques de l’Ecole Normale Supérieure (3), 38, 341-388 (1921) · JFM 48.0200.02
[36] Hahn, H., Reelle Funktionen (1932), Akademische Verlagsgesellschaft: Akademische Verlagsgesellschaft Leipzig · JFM 58.0242.05
[37] Hausdorff, F., Grundzüge der Mengenlehre (1914), Veit: Veit Leipzig · JFM 45.0123.01
[38] Hausdorff, F., Mengenlehre (1927), de Gruyter: de Gruyter Berlin · JFM 53.0169.01
[39] Hawkins, T., Lebesgue’s Theory of Integration. Its Origins and Development (1979), Chelsea: Chelsea New York
[40] Hille, E., Analytic Function Theory, vol. I (1959), Ginn: Ginn Boston · Zbl 0088.05204
[41] Hobson, E. W., The Theory of Functions of a Real Variable and the Theory of Fourier’s Series (1907), Cambridge University Press: Cambridge University Press Cambridge, UK · JFM 38.0414.01
[42] Hobson, E. W., The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, vol. 1 (1927), Cambridge University Press: Cambridge University Press Cambridge, UK · JFM 53.0226.01
[43] Hurwitz, A., Über die Entwickelung der allgemeinen Theorie der analytischen Funktionen in neuer Zeit, (Rudio, F., Verhandlungen des ersten internationalen Mathematiker-Kongresses in Zürich vom 9. bis 11. August 1897 (1898), Teubner: Teubner Leipzig), 91-112
[44] (James, I. M., History of Topology (1999), North-Holland: North-Holland Amsterdam) · Zbl 0922.54003
[45] Jordan, C., Remarques sur les intégrales définis, Journal de mathématiques pures et appliquées (4), 8, 69-99 (1892) · JFM 24.0261.01
[46] Jordan, C., Cours d’analyse de l’Ecole Polytechnique, vol. 1 (1893), Gauthier-Villars: Gauthier-Villars Paris · JFM 24.0247.03
[47] Kelley, J. L., General Topology (1955), Van Nostrand: Van Nostrand Princeton, NJ · Zbl 0066.16604
[48] Kuratowski, K., Sur l’opération \(\overline{A}\) de l’analysis situs, Fundamenta Mathematicae, 3, 182-199 (1922) · JFM 48.0210.04
[49] Kuratowski, K., Topologie I. Espaces métrisables, espaces complets (1933), Garasiński: Garasiński Warsaw · JFM 59.0563.02
[50] Lebesgue, H., Intégrale, longueur, aire, Annali di matematica pura ed applicata (3), 7, 231-359 (1902), Pagination follows the original printing of the dissertation, pp. 1-129 · JFM 33.0307.02
[51] Lebesgue, H., Sur les fonctions représentables analytiquement, Journal de mathématiques pures et appliqués, 60, 139-216 (1905) · JFM 36.0453.02
[52] Lebesgue, H., Leçons sur l’intégration et la recherche des fonctions primitives (1928), Gauthier-Villars: Gauthier-Villars Paris · JFM 54.0257.01
[53] Lefschetz, S., L’analysis situs et la géométrie algébrique (1924), Gauthier-Villars: Gauthier-Villars Paris · JFM 50.0663.01
[54] Lefschetz, S., Topology (1930), American Mathematical Society: American Mathematical Society New York · JFM 56.0491.08
[55] Lefschetz, S., Algebraic Topology (1942), American Mathematical Society: American Mathematical Society New York · Zbl 0061.39302
[56] Lennes, N. J., Curves in non-metrical analysis situs (abstract), Bulletin of the American Mathematical Society, 12, 284 (1906)
[57] Lennes, N. J., Curves in non-metrical analysis situs with an application in the calculus of variations, American Journal of Mathematics, 33, 287-326 (1911) · JFM 42.0399.01
[58] Listing, J. B., Vorstudien zur Topologie, Göttinger Studien, 2, 811-875 (1847)
[59] Marsden, J. E.; Hoffman, M. J., Basic Complex Analysis (1987), Freeman: Freeman New York · Zbl 0644.30001
[60] Mittag-Leffler, G., Sur la représentation analytique des fonctions monogènes uniformes d’une variable indépendante, Acta Mathematica, 4, 1-79 (1884) · JFM 16.0351.01
[61] Moore, G. H., Zermelo’s Axiom of Choice: Its Origins, Development and Influence (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0497.01005
[62] Moore, G. H., The axiomatization of linear algebra: 1875-1940, Historia Mathematica, 22, 262-303 (1995) · Zbl 0835.01007
[63] Moore, G. H., Historians and philosophers of logic: Are they compatible? The Bolzano-Weierstrass theorem as a case study, History and Philosophy of Logic, 20, 169-180 (2000) · Zbl 1052.03502
[64] Moore, G. H., The evolution of the concept of homeomorphism, Historia Mathematica, 34, 333-343 (2007) · Zbl 1129.01006
[65] Newman, M. H.A., Elements of the Topology of Plane Sets of Points (1939), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0021.06704
[66] Newman, M. H.A., Elements of the Topology of Plane Sets of Points (1951), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0045.44003
[67] Osgood, W. F., Allgemeine Theorie der analytischen Funktionen a) einer und b) mehrerer komplexen Grössen, Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, II B, 1, 1-45 (1901)
[68] Osgood, W. F., Lehrbuch der Funktionentheorie (1928), Teubner: Teubner Leipzig · Zbl 0005.29904
[69] Peano, G., Applicazione geometriche del calcolo infinitesimale (1887), Fratelli Bocca: Fratelli Bocca Turin · JFM 19.0248.01
[70] Pincherle, S., Saggio di una introduzione alla teoria delle funzioni analitiche secondo i principi del prof. C. Weierstrass, Giornale di matematiche, 18, 178-254 (1880), 317-357 · JFM 12.0307.01
[71] Poincaré, H., Mémoires sur les groupes kleiniens, Acta Mathematica, 3, 49-92 (1883) · JFM 15.0348.02
[72] Pont, J. C., La topologie algébrique, des origines à Poincaré (1974), Presses Universitaires de France: Presses Universitaires de France Paris · Zbl 0319.55001
[73] Riesz, F., Sur un théorème de M. Borel, Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Paris, 140, 224-226 (1905) · JFM 36.0103.01
[74] Schoenflies, A., Die Entwickelung der Lehre von den Punktmannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung, 8, ii, 1-251 (1900) · JFM 31.0070.08
[75] Schoenflies, A., Beiträge zur Theorie der Punktmengen, I, Mathematische Annalen, 58, 195-238 (1904) · JFM 34.0074.03
[76] Seifert, H.; Threlfall, W., Lehrbuch der Topologie (1934), Teubner: Teubner Leipzig · Zbl 0009.08601
[77] Sierpiński, W., La notion de dérivée comme base d’une théorie des ensembles abstraits, Mathematische Annalen, 97, 321-337 (1926) · JFM 52.0582.02
[78] Sierpiński, W., Topologia ogólna (1928), Kasa im. Mianowskiego: Kasa im. Mianowskiego Warsaw
[79] Sierpiński, W., Introduction to General Topology (1934), University of Toronto: University of Toronto Toronto · JFM 60.0502.01
[80] Tannery, J., Review of Cantor, Bulletin des sciences mathématiques et astronomiques (2), 2, 162-171 (1884)
[81] Tannery, J., Introduction à la théorie des fonctions d’une variable (1886), Hermann: Hermann Paris · JFM 18.0328.02
[82] Tannery, J., Review of Peano, Bulletin des sciences mathématiques et astronomiques (2), 11, 237-239 (1887)
[83] Tietze, H., Beiträge zur allgemeinen Topologie. I. Axiome für verschiedene Fassungen des Umgebungsbegriffs, Mathematische Annalen, 88, 290-312 (1923) · JFM 49.0397.02
[84] Veblen, O., The Cambridge Colloquium 1916. Part II: Analysis Situs (1922), American Mathematical Society: American Mathematical Society New York · JFM 57.0712.01
[85] Weierstrass, C., 1865-1866. Prinzipien der Theorie der analytischen Functionen. Unpublished lecture notes taken by Moritz Pasch at the University of Berlin. Kept in Pasch’s Nachlass; Weierstrass, C., 1865-1866. Prinzipien der Theorie der analytischen Functionen. Unpublished lecture notes taken by Moritz Pasch at the University of Berlin. Kept in Pasch’s Nachlass
[86] Weierstrass, C., Einführung in die Theorien der analytischen Functionen, Schriftenreihe des Mathematischen Instituts der Universität Münster (2), Heft 38 (1868/1986), Lecture notes taken by Wilhem Killing in 1868, published in 1986 in
[87] Weierstrass, C., 1874. Einleitung in die Theorien der analytischen Functionen. Lecture notes taken by G. Hettner at Berlin in the summer semester of 1874; Weierstrass, C., 1874. Einleitung in die Theorien der analytischen Functionen. Lecture notes taken by G. Hettner at Berlin in the summer semester of 1874
[88] Weierstrass, C., Zur Theorie der eindeutigen analytischen Functionen, Abhandlungen der Königl. Akademie der Wissenschaften zu Berlin (1876), Reprinted in [1895, 77-124]. Pagination agrees with the reprint · JFM 09.0283.04
[89] Weierstrass, C., Einleitung in die Theorie der analytischen Funktionen (1878/1988), Vieweg: Vieweg Braunschweig, Vorlesung Berlin 1878 in einer Mitschrift von Adolf Hurwitz (P. Ullrich, Ed.)
[90] Weierstrass, C., Zur Functionenlehre, Monatsbericht der Königl. Akademie der Wissenschaften zu Berlin, 719-743 (1880) · JFM 12.0310.01
[91] Weierstrass, C., Ausgewählte Kapitel aus der Funktionenlehre (1886/1988), Teubner: Teubner Leipzig, Vorlesung, gehalten in Berlin 1886. Mit der akademischen Antrittsrede, Berlin 1857, und drei weiteren Originalarbeiten von K. Weierstrass aus den Jahren 1870 bis 1880/86 (Reinhard Siegmund-Schultze, Ed.)
[92] Weierstrass, C., Mathematische Werke, vol. 2 (1895), Mayer & Müller: Mayer & Müller Berlin, (J. Knoblauch, G. Hettner, R. Rothe, Eds.)
[93] Whyburn, G., Analytic Topology (1942), American Mathematical Society: American Mathematical Society New York · Zbl 0061.39301
[94] Wilder, R. L., Evolution of the topological concept of “connected”, American Mathematical Monthly, 85, 720-726 (1978) · Zbl 0394.54009
[95] Young, W. H.; Young, G. C., The Theory of Sets of Points (1906), Cambridge University Press: Cambridge University Press Cambridge, UK · JFM 37.0070.01
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