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On the properties of domains and their boundaries inE n

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  1. The contents of this paper were presented to the American Mathematical Society in three parts: (1)A converse of the theorem regarding the separation of E 3 by a closed two-dimensional manifold of genus p (Bull. Amer. Math. Soc.36 (1930), p. 219, abstract no. 196); (2) second paper, under same title as (1) (ibid. Bull. Amer. Math.37 (1931), p. 519, abstract no. 236); (3)On Jordan continua that are the common boundaries of two or more domains in E n (ibid. Bull. Amer. Math.37, p. 525, abstract no. 258). The main results of (1) and (2) are embodied in Theorems 16 and 20 of the present paper, and were originally obtained by very different although much more complicated methods than those used in the present instance. It was because the author recognized that the results of (3) could be used to obtain simpler proofs of the theorems given in (1) and (2) that the above papers are here combined.

  2. Ergänzungsband, Jahresber. d. Deutsch. Math. Ver., 1908.

  3. AJordan continuum is a compact, connected and locally connected point set. Such continua are also variously termedcontinuous curves, Peano continua or simplylocally connected continua.

  4. See, for instance, G. Feigl,Geschichtliche Entwicklung der Topologie, Jahresb. d. Deutsch. Math.-Ver.37 (1928), pp. 273–286.

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  5. See, for instance, the latter part of chapter 5 of Schoenflies' work cited above.

  6. R. L. Wilder,A converse of the Jordan-Brouwer separation theorem in three dimensions, Trans. Amer. Math. Soc.32 (1930), pp. 632–657.

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  7. We shall use the “modulo 2” combinatorial topology throughout, since it seems sufficient to achieve the desired results. For an exposition of this see O. Veblen,Analysis Situs, Amer. Math. Soc. Coll. Pub.5, part II, 1931 (2nd ed.). A brief exposition which is sufficient for the purposes of the present paper will be found in the paper cited in 8) below.

  8. See my paperPoint sets in three and higher dimensions and their investigation by means of a unified analysis situs, Bull. Amer. Math. Soc.38 (1932), pp. 649–692.

  9. L. E. J. Brouwer,Beweis der Invarianz der geschlossenen Kurve, Math. Ann.72 (1912), pp. 422–425.

    Google Scholar 

  10. L. Vietoris,Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann.97 (1927), pp. 454–472. See also P. Alexandroff,Untersuchungen über Gestalt und Lage abgeschlossener Mengen beliebiger Dimension, Annals of Math.30 (1928–29), pp. 101–187.

    Google Scholar 

  11. L. E. J. Brouwer,Zur Analysis Situs, Math. Ann.68 (1910), pp. 422–444. See also the example of Wada referred to below.

    Google Scholar 

  12. C. Kuratowski,Sur les coupures irréductibles du plan, Fund. Math.6 (1924), pp. 130–145.

    Google Scholar 

  13. See the memoir of Alexandroff cited in 10) —.

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  14. It is well known that no Jordan continuum is an indecomposable continuum. This fact follows easily, for instance, from the property of local connectedness and a theorem of Knaster and Kuratowski. See B. Knaster and C. Kuratowski,Sur les ensembles connexes, Fund. Math.2 (1921), pp. 206–255, Th. VI.

    Google Scholar 

  15. See the memoir of K. Yoneyama,Theory of continuous sets of points, Tôhoku Math. Journ.12 (1917), p. 60.

    Google Scholar 

  16. IfD is a domain, byF (D) we mean the boundary, or frontier, ofD.

  17. L. E. J. Brouwer,Beweis des Jordanschen Kurvensatzes, Math. Ann.69 (1910), pp. 169–175.

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  18. IfG is an open subset ofE n we shall say that thei-cycles(i<n) bound uniformly in G or areuniformly homologous to zero in G ifp i(G)=0 and for every ɛ>0 there exists a δ>0 such that everyi-cycle ofG of diameter <δ bounds a complex inG of diameter <ɛ.

  19. For any positive number δ the symbolS(P, δ) denotes the set of all points of space whose distance fromP is <δ. The boundary ofS(P, δ) we shall denote byF(P, δ).

  20. For the casen=2 see G. T. Whyburn,Concerning continua in the plane, Trans. Amer. Math. Soc.29 (1927), pp. 369–400, Theorem 19. For the converse of this lemma see R. L. Moore,Concerning the common boundary of two domains, Fund. Math.6 (1924), pp. 203–213, Theorem 8.

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  21. Loc. cit., For the casen=2 see, Theorem XXXVII, which holds inE n.

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  22. S. Mazurkiewicz,Sur un ensemble G δ,punctiforme, qui n'est pas homéomorphe avec aucun ensemble linéaire, Fund. Math.1 (1920), pp. 61–81, Theorem I.

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  23. See P. Alexandroff, loc. cit., p. 154.

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  24. AnM-domain is a connected open subset ofM.

  25. R. L. Wilder,On the imbedding of subsets of a metric space in Jordan continua, Fund. Math.19 (1932), pp. 45–64.

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  26. J. W. Alexander,A proof and extension of the Jordan-Brouwer separation theorem, Trans. Amer. Math. Soc.23 (1922), pp. 333–349, CorollaryW i. For future reference we note that TheoremY of this paper is what is known as the Alexander Duality Theorem.

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  27. See K. Menger,Dimensionstheorie, p. 295.

  28. By well-known duality relations. See P. Alexandroff,Über die Dualität zwischen den Zusammenhangszahlen einer abgeschlossenen Menge und des zu ihr komplementären Raumes, Gött. Nachr. 1927, p. 323; F. Frankl,Topologische Beziehungen in sich kompakter Teilmengen euklidischer Räume zu ihren Komplementen sowie Anwendung auf die Prim-Enden-Theorie, Wien. Akad. d. Wiss., Math.-Naturw. Kl., Sitz., Abt. 2 A,136 (1927), pp. 689–699; P. Alexandroff, loc. cit,Über die Dualität zwischen den Zusammenhangszahlen einer abgeschlossenen Menge und des zu ihr komplementären Raumes, Gött. Nachr. 1927, pp. 156ff; S. Lefschetz,Closed point sets on a manifold, Annals of Math.29 (1928), pp. 232–254.

  29. Proved forn=3 and suggested for the general case by Urysohn; see P. Urysohn,Mémoire sur les multiplicités Cantoriennes, Fund. Math.7 (1925), pp. 30–137 (especially p. 123) and ibid. Fund. Math.8 (1926), pp. 225–356 (especially pp. 311–313). Proved for the general case by P. Alexandroff, loc. cit.Über die Dualität zwischen den Zusammenhangszahlen einer abgeschlossenen Menge und des zu ihr komplementären Raumes, Gött. Nachr. 1927, (p. 154).

    Google Scholar 

  30. See P. Alexandroff, loc. cit.Über die Dualität zwischen den Zusammenhangszahlen einer abgeschlossenen Menge und des zu ihr komplementären Raumes, Gött. Nachr. 1927, (p. 154).

  31. A Jordan continuumM is calledcyclicly connected if every two points ofM lie on a simple closed curve ofM. Every two points of a locally connected continuum having no cut point lie on a simple closed curve of that continuum; see G. T. Whyburn,On the cyclic connectivity theorem, Bull. Amer. Math. Soc.37 (1931), pp. 429–433, and earlier papers by the same author and W. L. Ayres referred to therein.

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  32. We recall the fact that a continuum which is a common boundary of two or more domains inE n is not necessarily a closed cantorian manifold, since it may have other complementary domains of which it is not itself the boundary, in the sense that their boundaries areproper subsets of the continuum.

  33. Fork=0, Theorem 3 may be applied.

  34. L. Zippin,On continuous curves and the Jordan Curve Theorem, Amer. Journ. Math.52 (1930), pp. 331–350.

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  35. J. W. Alexander,loc. cit., TheoremX i. It is to be noted that the numbersp i are uniformlyless by unity than the numbersR i employed in Alexander's paper.

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  36. See R. L. MooreOn the relations of a continuous curve to its complementary domains in space of three dimensions, Proc. Nat. Acad. Sci. 8 (1922), pp. 33–38; in this paper will also be found some interesting examples of continuous curves in three dimensions; also see my paper referred to in6), p. 644, II, a slight alteration in the last paragraph of which yields a proof of this for the case where there may be other complementary domains.

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  37. See R. L. Moore,Concerning continuous curves in the plane, Math. Zeitschr.15 (1922), pp. 254–260, Th. 5, and C. Kuratowski, loc. cit.Sur les coupures irréductibles du plan, Fund. Math.6 (1924), pp. 130–145, Th. VII.

    Google Scholar 

  38. See G. T. Whyburn,Concerning accessibility in the plane and regular accessibility in n dimensions. Bull. Amer. Math. Soc.34 (1928), pp. 504–510.

    Google Scholar 

  39. That is, for every pointP ofK there exists ans>0 such that ifC is the component ofK determined byP inS (P, ε) andF⊂C, thenC−F is connected.

  40. LetC be defined as in39) Then asC has no cut-point it is cyclicly connected [See footnote31)].—

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  41. G. T. Whyburn,Continuous curves without local separating points, Amer. Journ. Math.53 (1931), pp. 163–166.

    Google Scholar 

  42. See T. Rado, Über den Begriff der Riemannschen Fläche, Acta Litt. ac Scient. (Szeged)2 (1925), pp. 101–121.

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  43. That is, eachE 1i is a simple arc.

  44. See L. Pontrjagin,Zum Alexanderschen Dualitätssatz, Göttinger Nachr. (1927), pp. 315–322, Th. 3.

  45. Hereafter, ifK is a complex, we shall denote by (K) the point set consisting of all points inK.

  46. See R. L. Moore,Concerning continuous curves in the plane,loc. cit., Th.1; the proof of this theorem is valid when the imbedding space is anyE n.

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  47. Loc. cit. L. Pontrjagin,Zum Alexanderschen Dualitätssatz, Göttinger Nachr. (1927), pp. 315–322 Theorem II.

  48. For a summary of the possible types of closed 2-dimensional manifolds, see Veblen,Analysis situs, pp. 50–51. (Note that Veblen's numbersR i are uniformly greater by unity than the numbers p i which we are using).

  49. This follows readily from the fact thatC is anopen subset ofK.

  50. For the uniform local connectedness of the domains complementary to a manifold, see my paper referred to in6).

  51. That is, it is not necessary to assume thatE 8−K consists of exactly two domains. It might be noted here that in the plane case Schoenflies assumed (Über einen grundlegenden Satz der Analysis Situs, Göttinger Nachr. 1902, p. 185) thatp 0 (E2—K)=1, and it was later shown by P. M. Swingle (An unnecessary condition in two theorems of Analysis Situs, Bull. Amer. Math. Soc.34 (1928), pp. 607–618) that in the presence of Schoenflies' other conditions (which included the condition that p0 (E2—K)≧1) this condition was unnecessary. I should also like to call attention to the fact that whereas in the proof of the converse theorem for simple closed surfaces given in my paper referred to in6) it was felt necessary, in order to associate the combinatorial conditions imposed on the complement of the setK with the topological properties ofK, to introduce a new definition of the simple closed surface in terms of what might be calledproperties in the large, we have succeeded in the proof of Theorem 16 in associating the combinatorial properties ofE 3—K directly with thelocal topological properties ofK.

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  52. L. Pontrjagin, loc. cit.,Zum Alexanderschen Dualitätssatz, Göttingen Nachr. (1927), pp. 315–322, Theorem II.

  53. Since a (compact) common boundary of two uniformly locally connected domains is a Jordan continuum, it might occur to the reader to ask, is a locally connected 2-dimensional closed cantorian manifoldK such thatp 1 (K) is finite necessarily a closed 2-dimensional (combinatorial) manifold? That such is not the case is shown immediately by the surfaceS 1 of Theorem 1. However, a locally connected 1-dimensional cantorian manifold is a 1-dimensional (combinatorial) manifold (=simple closed curve), as shown in my paperOn the linking of Jordan continua by (n−2)-cycles, to appear soon in the Annals of Math.

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Wilder, R.L. On the properties of domains and their boundaries inE n . Math. Ann. 109, 273–306 (1934). https://doi.org/10.1007/BF01449139

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