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Cantor–Schoenflies Topology

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Book cover L.E.J. Brouwer – Topologist, Intuitionist, Philosopher

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Abstract

When Brouwer continued his investigations into Hilbert 5, he discovered that his main topology source, Schoenflies Bericht, was far from correct. He set himself to straighten out the defective parts; the best known fall out of this research was his work on indecomposable continua, with the spectacular example of three domains with one common boundary. The chapter also contains the story of Brouwer’s research on fixed points on the sphere and his translation theorem (on fixed point free continuous maps of the plane onto itself). He simultaneously produced a number of papers on vector field on surfaces. The best known result was the hairy ball theorem: a continuous vector field on a sphere must be zero or infinite at at least one point.

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Notes

  1. 1.

    Such a transformation f can be reversed. To be precise: there is another transformation g, so that f followed by g, and g followed by f take points to themselves.

  2. 2.

    Alexander Soifer pointed out an omission in the first formulation.

  3. 3.

    Actually a misnomer, cf. Freudenthal (1954).

  4. 4.

    The problem was eventually solved in 1952 in the affirmative by A.M. Gleason, D. Montgomery and L. Zippin. As a consequence one may now define a Lie group as a topological group over a locally Euclidean manifold.

  5. 5.

    Brouwer (1908a, 1909b).

  6. 6.

    Die Theorie der endlichen continuierlichen Gruppen unabhängig von den Axiomen von Lie.

  7. 7.

    Brouwer (1908c).

  8. 8.

    Freudenthal, CW II, p. 101.

  9. 9.

    On Analysis Situs (Brouwer 1910e).

  10. 10.

    CW II, p. 117.

  11. 11.

    Antoine (1921).

  12. 12.

    Alexander (1924).

  13. 13.

    Brouwer to Scheltema 9 November 1909.

  14. 14.

    45 years, actually.

  15. 15.

    See Brouwer (1928c).

  16. 16.

    Taussky to Van Dalen, 1991.

  17. 17.

    Hilbert (1902, 1909) and later editions.

  18. 18.

    Brouwer to Hilbert, 28 October 1909, reproduced in CW II, p. 102 ff.

  19. 19.

    Brouwer to Scheltema, 9 November 1909.

  20. 20.

    The only hint that could throw doubt on the good relationship is a letter of Bernays to Freudenthal, relating to the above mentioned letter of Brouwer to Hilbert. In this letter Bernays more or less expressed his surprise that Hilbert and Brouwer were still good friends at that time. There are, in my opinion, no historical facts that would support this. But it is, of course, possible that Hilbert expressed in private conversation criticism of Brouwer.

  21. 21.

    Brouwer (1909c).

  22. 22.

    The volume for 1909 appeared in 1912.

  23. 23.

    In Freudenthal’s words: ‘Engel, […] who could not grasp a group except in its analytic setting, and Brouwer, who had shaken off the algorithmic yoke and from his conceptional viewpoint could not comprehend his correspondent’s difficulties. Manifolds and one-to-one mappings as substrate and action of Lie groups instead of Cartesian space and many-valued mappings was indeed a great step forward, though for older contemporaries of Brouwer it was too much’ (CW II, p. 142).

  24. 24.

    Brouwer to Engel, 21 January 1912.

  25. 25.

    That is the excessive restrictions.

  26. 26.

    Engel to Brouwer, 28 January 1912, CW II, p. 144.

  27. 27.

    Brouwer to Engel, 6 March 1912.

  28. 28.

    Die Theorie der endlichen kontinuierlichen Gruppen, unabhängig von den Axiomen von Lie, II., Brouwer (1910b).

  29. 29.

    Het Nederlandsch Natuur- en Geneeskundig Congres.

  30. 30.

    Characterisation of the Euclidean and non-Euclidean motion groups in R n , CW II, Brouwer (1909d) (originally in Dutch).

  31. 31.

    Brouwer (1911f).

  32. 32.

    Brouwer (1909e).

  33. 33.

    Brouwer (1909e), pp. 10, 11.

  34. 34.

    Popularly speaking one fixes the orientation on a surface by indicating a direction on a little circle, the orientation at other places is determined by shifting this circle all over the surface. If this is possible, that is if one never can get via different routes two circles circulating in opposite directions around one point, the surface is called orientable, and it has an orientation. The sphere is an example of such a surface. The Möbius strip is an example of a one-sided non-orientable surface (take a rectangular strip of paper, twist it and glue the ends together).

  35. 35.

    When the publisher, at the end of the series, mixed up the numbering, the numbers 7 and 8 were erroneously published as 6 and 7. Brouwer patiently corrected the numbering by hand in the off-prints.

  36. 36.

    Die Entwicklung der Lehre von den Punktmannigfaltigkeiten II, 1908.

  37. 37.

    Brouwer (1912h), p. 360.

  38. 38.

    Über eineindeutige, stetige Transformationen von Flächen in sich, Brouwer (1910g).

  39. 39.

    Brouwer (1909g).

  40. 40.

    Cf. Brouwer (1910e).

  41. 41.

    (Brouwer’s note) Already the property of p. 288 that the transformation domain constructed in the way indicated there, determines at most two residual domains, vanishes for some domains, incompatible with the Schoenflies theory.

  42. 42.

    Brouwer (1912b).

  43. 43.

    Brouwer (1919n).

  44. 44.

    Cf. CW II, p. 220, and Freudenthal’s comments on p. 219.

  45. 45.

    Brouwer (1909f).

  46. 46.

    Peano (1890). Brouwer also cited the simplification of Arzela (1896).

  47. 47.

    Poincaré (1881). Translation by Freudenthal. CW II, p. 282.

  48. 48.

    Cf. p. 199.

  49. 49.

    Cf. Johnson (1981), p. 154. The letter from Hadamard was undated, but it seems likely that it was answered promptly by Brouwer.

  50. 50.

    Brouwer (1909f), p. 856.

  51. 51.

    CW II, p. 282.

  52. 52.

    Brouwer (1911c), p. 112.

  53. 53.

    Brouwer (1910c, 1910f).

  54. 54.

    Brouwer (1910c).

  55. 55.

    Cf. Freudenthal, CW II, p. 302.

  56. 56.

    Cf. Johnson (1981), p. 153.

  57. 57.

    Brouwer (1909e).

  58. 58.

    Hadamard to Brouwer 24 December 1909.

  59. 59.

    Brouwer (1910f), CW II, p. 314.

  60. 60.

    Section 3 of Brouwer (1910f). See also Hadamard’s acknowledgement in his appendix to the second edition of Tannery’s Introduction à la théorie des fonctions II (dealing with topological applications of the Kronecker index), Hadamard (1910).

  61. 61.

    Brouwer (1915).

  62. 62.

    Krystall Systeme und Krystallstruktur (1891); a new edition was published in 1984!

  63. 63.

    There were other texts, for example, Young and Young (1906) but they had only a modest influence.

  64. 64.

    Zermelo (1908), Schoenflies (1905).

  65. 65.

    Schoenflies (1903, 1904, 1906).

  66. 66.

    Brouwer (1909b).

  67. 67.

    Over vlakke krommen en vlakke gebieden.

  68. 68.

    Brouwer to Hilbert, 14 May 1909.

  69. 69.

    Brouwer (1910e).

  70. 70.

    A domain is a connected open set. Schoenflies (1908), p. 112.

  71. 71.

    A continuum is a compact connected set with at least two points.

  72. 72.

    Yoneyama (1917).

  73. 73.

    Brouwer (1912i).

  74. 74.

    Bemerkung zu dem vorstehenden Aufsatz des Herrn L.E.J. Brouwer.

  75. 75.

    Cf. p. 226.

  76. 76.

    Brouwer (1910a).

  77. 77.

    Phragmén (1885), cf. Freudenthal, CW II, p. 383.

  78. 78.

    Cf. Dieudonné (1989), p. 207.

  79. 79.

    Proved in Brouwer (1912e, 1912i).

  80. 80.

    Proved in Brouwer (1911e).

  81. 81.

    Brouwer (1919d, 1919e, 1919j, 1919m).

  82. 82.

    Scheveningen, September 1909, cf. p. 125.

  83. 83.

    Brouwer (1910d, 1911a).

  84. 84.

    Poincaré (1912).

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  1. Department of Philosophy, Utrecht University, Utrecht, Netherlands

    Dirk van Dalen

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van Dalen, D. (2013). Cantor–Schoenflies Topology. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_4

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