AN EPISTEMOLOGICAL ANALYSIS OF COMPACTNESS IN ANALYSIS AND TOPOLOGY
Main Article Content
Abstract
Compactness in metric and topological spaces is one of the basic concepts of Topology. It is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Many of the concepts in the Topology as well as in the metric space are based on this concept. This paper presents an epistemological analysis that clarifies the emergence and development of the concept of compactness and determines the epistemological characteristics of these two knowledge objects.
Keywords
compactness, epistemological characteristic, metric space, topogical space, epistemological analysis
Article Details
References
Alexandroff, P. (1924). Ueber die Metrisation der im Kleinen kompakten topologischen Râume. Mathematische Annalen. Berlin, Julius Springer, 92.
Alexandroff, P., & Urysohn, P. (1924). Zur Théorie der topologischen Râume. Mathematische Annalen, 92.
Alexandroff, P. (1925). Zur Begriindung der n-dimensionalen mengentheo-retischen Topologie. Mathematische Annalen, (94).
Bolzano, B. (1817). Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewa ̈hren, wenigstens eine reelle Wurzel der Gleichung liege. Prague.
Brousseau, G., (1983). “Les obstacles épistémologiques et les problèmes en mathématiques”, Recherches en Didactique des Mathématiques, 4(2), 141-163.
Bourbaki, N. (1951). Structures topologiques. Structures uniformes. Paris, Hermann & Cie., 91-129.
Cartan, H. (1937a). Théorie des filtres. Comptes rendus de l’Académie des Sciences. Paris, 205.
Cartan, H. (1937b). Filtres et ultrafiltres. Comptes rendus de l'Académie des Sciences, Paris, 205.
Chandler, R., & Faulkner, G. (2001). Hausdorff compactifications: A retrospective, in Handbook of the History of General Topology. Edited by C. E. Aull, R. Lowen. Springer Verlag, Berlin.
Cousin, P. (1895). Sur les fonctions de n variables complexes. Acta Math., (19), 1-61.
Dang, D. A. (2001). Introduction of Analysis. Ed. Education.
Dugac, P. (1989). Sur la correspondance de Borel et le theoreme de Dirichlet-Heine Weierstrass-Borel-Schoenflies-Lebesgue, Arch. Inter. Hist. Sci., 39(122), 69-110.
Ferrari, P. L. (2003). Abstraction in mathematics. Philosophical Transactions: Biological Sciences, 358(1435), The Abstraction Paths: From Experience to Concept, The Royal Society, 1225-1230.
Fréchet, M. (1904). Généralisation d’un théorème de Weierstrass. C. R. Acad. Sci. Paris 139, 848-850.
Fréchet, M. (1906). Sur quelques points du calcul fonctionnel. Rend. Palermo 22, 1-74.
Fréchet, M. (1917). Sur la notion de voisinage dans les ensembles abstraits, Comptes rendus de l'Académie des Sciences, 165, 359-360.
Fréchet, M. (1928). Les espaces abstraits. Paris, Gauthier-Villars.
Hausdorff, F. (1914). Grundzüge der Mengenlehre. Verlag von Veit, Leipzig.
Jahnke, H. N. (2016). A history of Analysis. American Mathematical Society.
Katz, V. J. (2009). A History of Mathematices-An Introduction. 3rd Edition, Pearson Education, Inc.
Kline, K. (1972). Mathematical Thought: From Ancient to Modern Times. Oxford University Press.
Le, V. T. (2003). A new perspective on the process of teaching the concept of mathematics. Journal of Education, (64), Ha noi.
Le, T. H. C, & Le, V. T. (2003). The role of epistemological analysis of History of Mathematics in research and practice of mathematics teaching and learning. Research theme at the Ministry Level, Code B2001-23-2.
Marquis, J. P. (2015). Mathematical abstraction, conceptual variation and identity. In Peter Schroeder-Heister, Gerhard Heinzmann, Wilfred Hodges & Pierre Edouard Bour, Logic, Methodology and Philosophy of Science, 229-321. London: College Publications.
Nguyen, A. Q., & Nguyen, T. V. K. (2017). An epistemological analysis in teaching advanced algebra: the case of group. 6th International Seminar on Didactic Mathematics, Ho Chi Minh City University of Education.
Nguyen, A. Q., & Nguyen, T. T. T. (2017). The contributions of epistemological analysis for teaching linear algebra: the case of vector space. 6th International Seminar on Didactic Mathematics, Ho Chi Minh City University of Education.
Pier, J. P. (1961). Genèse et évolution de l’idée de compact. Revue d’histoire des sciences et de leurs applications. Tome 14, (2), 69-179.
Pier, J. P. (1984). Historique de la notion de compacité. Historica Mathematica 7, 425-443.
Raman-Sundström, M. (2015). A Pedagogical History of Compactness. The American Mathematical Monthly, 122(7), 619-635.
Sutherland, W. A. (2009). Introduction to Metric and Topological Spaces. Second Edition, Oxford University Press.
Taylor, A. (1982). A study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals. Arch. Hist. Exact Sci, 27(3), 233-295.
Taylor, A. E. (1985). A study of Maurice Fréchet: II. Mainly about his work on general topology, 1909-1928, Arch. Hist. Exact Sci., 34(3), 279-380.
Weil, A. A. (1937). Sur les espaces à structure uniforme et sur la topologie générale. Paris,
Hermann & Cie.
Alexandroff, P., & Urysohn, P. (1924). Zur Théorie der topologischen Râume. Mathematische Annalen, 92.
Alexandroff, P. (1925). Zur Begriindung der n-dimensionalen mengentheo-retischen Topologie. Mathematische Annalen, (94).
Bolzano, B. (1817). Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewa ̈hren, wenigstens eine reelle Wurzel der Gleichung liege. Prague.
Brousseau, G., (1983). “Les obstacles épistémologiques et les problèmes en mathématiques”, Recherches en Didactique des Mathématiques, 4(2), 141-163.
Bourbaki, N. (1951). Structures topologiques. Structures uniformes. Paris, Hermann & Cie., 91-129.
Cartan, H. (1937a). Théorie des filtres. Comptes rendus de l’Académie des Sciences. Paris, 205.
Cartan, H. (1937b). Filtres et ultrafiltres. Comptes rendus de l'Académie des Sciences, Paris, 205.
Chandler, R., & Faulkner, G. (2001). Hausdorff compactifications: A retrospective, in Handbook of the History of General Topology. Edited by C. E. Aull, R. Lowen. Springer Verlag, Berlin.
Cousin, P. (1895). Sur les fonctions de n variables complexes. Acta Math., (19), 1-61.
Dang, D. A. (2001). Introduction of Analysis. Ed. Education.
Dugac, P. (1989). Sur la correspondance de Borel et le theoreme de Dirichlet-Heine Weierstrass-Borel-Schoenflies-Lebesgue, Arch. Inter. Hist. Sci., 39(122), 69-110.
Ferrari, P. L. (2003). Abstraction in mathematics. Philosophical Transactions: Biological Sciences, 358(1435), The Abstraction Paths: From Experience to Concept, The Royal Society, 1225-1230.
Fréchet, M. (1904). Généralisation d’un théorème de Weierstrass. C. R. Acad. Sci. Paris 139, 848-850.
Fréchet, M. (1906). Sur quelques points du calcul fonctionnel. Rend. Palermo 22, 1-74.
Fréchet, M. (1917). Sur la notion de voisinage dans les ensembles abstraits, Comptes rendus de l'Académie des Sciences, 165, 359-360.
Fréchet, M. (1928). Les espaces abstraits. Paris, Gauthier-Villars.
Hausdorff, F. (1914). Grundzüge der Mengenlehre. Verlag von Veit, Leipzig.
Jahnke, H. N. (2016). A history of Analysis. American Mathematical Society.
Katz, V. J. (2009). A History of Mathematices-An Introduction. 3rd Edition, Pearson Education, Inc.
Kline, K. (1972). Mathematical Thought: From Ancient to Modern Times. Oxford University Press.
Le, V. T. (2003). A new perspective on the process of teaching the concept of mathematics. Journal of Education, (64), Ha noi.
Le, T. H. C, & Le, V. T. (2003). The role of epistemological analysis of History of Mathematics in research and practice of mathematics teaching and learning. Research theme at the Ministry Level, Code B2001-23-2.
Marquis, J. P. (2015). Mathematical abstraction, conceptual variation and identity. In Peter Schroeder-Heister, Gerhard Heinzmann, Wilfred Hodges & Pierre Edouard Bour, Logic, Methodology and Philosophy of Science, 229-321. London: College Publications.
Nguyen, A. Q., & Nguyen, T. V. K. (2017). An epistemological analysis in teaching advanced algebra: the case of group. 6th International Seminar on Didactic Mathematics, Ho Chi Minh City University of Education.
Nguyen, A. Q., & Nguyen, T. T. T. (2017). The contributions of epistemological analysis for teaching linear algebra: the case of vector space. 6th International Seminar on Didactic Mathematics, Ho Chi Minh City University of Education.
Pier, J. P. (1961). Genèse et évolution de l’idée de compact. Revue d’histoire des sciences et de leurs applications. Tome 14, (2), 69-179.
Pier, J. P. (1984). Historique de la notion de compacité. Historica Mathematica 7, 425-443.
Raman-Sundström, M. (2015). A Pedagogical History of Compactness. The American Mathematical Monthly, 122(7), 619-635.
Sutherland, W. A. (2009). Introduction to Metric and Topological Spaces. Second Edition, Oxford University Press.
Taylor, A. (1982). A study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals. Arch. Hist. Exact Sci, 27(3), 233-295.
Taylor, A. E. (1985). A study of Maurice Fréchet: II. Mainly about his work on general topology, 1909-1928, Arch. Hist. Exact Sci., 34(3), 279-380.
Weil, A. A. (1937). Sur les espaces à structure uniforme et sur la topologie générale. Paris,
Hermann & Cie.