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Concerning points of continuous curves defined by certain im kleinen properties

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Literatur

  1. The pointP of a connected setM is said to be a cut point ofM providedM−P is not connected. The notion of an im kleinen cut point of a continuum is contained implicitly in the works of P. Urysohn and R. L. Moore, and is closely approximated in that of R. G. Lubben and C. Zarankiewicz. Cf. P. Urysohn, Über im kleinen zusammenhängende Kontinua, Math. Annalen98 (1927), S. 296–308; [Urysohn uses the terms “unvermeidbar” (unavoidable) and “vermeidbar” (avoidable) to designate im kleinen cut points and non-im kleinen cut points, respectively]; R. L. Moore, Concerning Triods in the Plane and the Junction Points of Plane Continua, Proc. Ntl. Acad. of Sci.14 (1928), pp. 85–88; R. G. Lubben, Concerning Connectedness near a Point Set; and C. Zarankiewicz, Sur les points de division dans les ensembles connexes, Fund. Math.9 (1927), see proof of Theorem 14.

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  2. Cf. K. Menger, Grundzüge einer Theorie der Kurven, Math. Annalen95 (1925), S. 272–306.

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  3. Cf. K. Menger,loc cit., and P. Urysohn, Comptes Rendus175 (1922), p. 481. Urysohn uses the term ‘index of a point’ instead of the therm ‘order of a point’.

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  4. That this definition is equivalent for the case of continuous curves to the Wilder definition [R. L. Wilder, Concerning Continuous Curves, Fund. Math.7 (1925), pp. 340–377] was shown by H. M. Gehman. See Concerning End Point of Continuous Curves and other Continua, Trans. Amer. Math. Soc.30 (1928). In my thesis I showed that this definition is (for plane continuous curvesM) equivalent to the following simple one:P is an endpoint ofM provided thatP is an interior point of no arc inM.Cf. Concerning Continua in the Plane, Trans. Amer. Math. Soc.29 (1927), pp. 369–400, Theorem 12. For the extension of this and other results frequently used later ton-space see W. L. Ayres, Concerning Continuous Curves in a Space ofn Dimensions, Amer. Journal of Math.

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  5. Cf. W. Sierpinski, Comptes Rendus160, p. 305. Sierpinski defines a ramification point ofM as a pointP such thatM contains 3 continuaK, L andN, such thatK·L=K·N=L·N=P. It follows from a result of Menger's (Fund. Math.10) that the definition here given and Sierpinski's definition are equivalent for Menger regular curves. Rutt [Bull. Amer. Math. Soc.33 (1927), p. 411 (abstract)] has shown them equivalent for all plane continuous curves. It appears likely that they are equivalent for continuous curves inn-space.

  6. Cf. my paper Cyclicly Connected Continuous Curves, Proc. Ntl. Acad. of Sci.13 (1927), pp. 31–38.

  7. Amer. Journ. of Math.50 (1928), pp. 167–194.

  8. The subsetR of a closed setM is said to be an open subset ofM providedM−R is either vacuous or closed.

  9. Bull. Amer. Math. Soc.34 (1928), pp. 349–360.

  10. See footnote 5)— and Theorem 3 above. AlthoughC itself is not necessarily a continuous curve, it follows by Theorem 3 thatC contains a continuous curveU such thatU⊃P and (M−U)′·P=0.

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  11. Concerning Continuous Curves in the Plane, Math. Zeitschr.15 (1922), Theorem 1. Also see footnote 12)—.

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  12. G. T. Whyburn, Concerning Collections of Cuttings of Connected Point Sets, Bull. Amer. Math. Soc.35 (1929), pp. 87–104.

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  13. Loc. cit. See footnote 11)Bull. Amer. Math. Soc.34 (1928), pp. 349–360.

  14. G. T. Whyburn, Concerning Continua in the Plane,loc. cit.; W. L. Ayres Concerning Continuous Curves and Correspondences, Ann. of Math.28 (1927), p. 396. For this theorem inn dimensions see W. L. Ayres, Concerning Continuous Curves in a Space ofn Dimensions,loc. cit. 5) Amer. Journal of Math.

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  15. See footnote 5)R. L. Wilder, Concerning Continuous Curves, Fund. Math.7 (1925), pp. 340–377.

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  16. C. Zarankiewicz,loc. cit., Sur les points de division dans les ensembles connexes, Fund. Math.9 (1927) Theorem 17.

  17. In this connection see also, R. L. Moore,loc. cit ref.2) Lemma 2.

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  18. Cf. W. H. Young, Leipz. Ber.55 (1903), S. 287.

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  19. G. T. Whyburn, Cyclicly Connected Continuous Curves,loc. cit..

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  20. Cf. G. T. Whyburn, Concerning Collections of Cuttings of Connected Point Sets,loc. cit..

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  21. R. L. Moore,loc. cit. see footnote 12).

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  22. K. Menger,loc. cit., see footnote 3).

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  23. See footnote 21)G. T. Whyburn, Cyclicly Connected Continuous Curves, Proc. Ntl. Acad. of Sci.13, (1927), pp. 31–38.

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  24. For a given ε>0, the pointP ofM is said to be, ε-separated inM by a setA providedM−A=M p +M 0 whereM p andM 0 are mutually separated sets andM p contains the pointP and is of diameter<ε.Cf. P. Urysohn, Comptes Rendus175 (1922), p. 481. Urysohn's definition differs from the one just given in that it requires that the setM p +A be of diameter <ε.

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  25. K. Menger (Zur allgemeinen Kurventheorie, Fund. Math.10) has proposed the question as to whether or not every regular curve has the property mentioned in the last sentence of this theorem.

  26. Grundzüge einer Theorie der Kurven,loc. cit. Math. Annalen95 (1925), S. 272–306.

  27. Cf. Kuratowski and Knaster, Remark on a Theorem of R. L. Moore, Proc. Ntl. Acad. of Sci.13 (1927); G. T. Whyburn, On the Separation of Connected Point Sets, Bull. Amer. Math. Soc.33 (1927), p. 388 (abstract).

  28. G. T. Whyburn, Concerning Menger Regular Curves, Fund. Math.12, Theorem 2.

  29. K. Menger,loc. cit., see ref. 27) (Zur allgemeinen Kurventheorie, Fund. Math.10).

  30. Fund. Math. 12, see proof of Theorem 2. For a statement of practically the same theorem see a forthcoming paper of W. L. Ayres entitled ‘Concerning Arc Curves and Basic Subsets of a Continuous Curve. Second Paper’.

  31. Cf. my paper Concerning the Structure of a Continuous Curve,loc. cit. 8) Amer. Journ. of Math.50 (1928), pp. 167–194.

  32. Cf. G. T. Whyburn, Concerning Irreducible Cuttings of Continua, Fund. Math.13, pp. 42–57.

  33. G. T. Whyburn,loc. cit., Concerning Irreducible Cuttings of Continua, Fund. Math.13, pp. 42–57. Theorem 7.

  34. G. T. Whyburn,loc. cit. Concerning Irreducible Cuttings of Continua, Fund. Math.13, pp. 42–57, Theorem 8.

  35. Loc. cit, Concerning Irreducible Cuttings of Continua, Fund. Math.13, pp. 42–57, Theorem 8.

  36. Concerning the Subsets of a Plane Continuous Curve, Annals of Math.27 (1925), pp. 29–46.

  37. Some Conditions Under Which a Continuum is a Continuous Curve, Ann. of Math.27 (1926), pp. 381–384, see Theorem 2.

  38. See reference 39) Concerning the Subsets of a Plane Continuous Curve, Annals of Math.27 (1925), pp. 29–46.

  39. Sur les Points de Division dans les Ensembles Connexes,loc. cit. Fund. Math.9 (1927);K is a continuum of convergence of a continuumM provided thatM−K contains a sequence of continua whose sequential limiting set isK.

  40. Concerning Certain Types of Continuous Curves, Proc. Ntl. Acad. of Sci12 (1926), pp. 761–767.

  41. Knaster and Kuratowski, Bull. Amer. Math. Soc.33 (1927), p. 106.

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  42. Loc. cit., see reference 43) Concerning Certain Types of Continuous Curves, Proc. Ntl. Acad. of Sci.12 (1926), pp. 761–767.

  43. Ann. of Math.29 (1928), pp. 399–411.

  44. That is, for each ε>0,R is the sum of a finite number of connected sets each of diameter <ε. See R. L. Moore, Fund. Math.3 (1922), p. 232.

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  45. For the former case, see my paper Concerning the Open Subsets of a Continuous Curve,loc. cit., Proc. Ntl. Acad. of Sci.13 (1927), pp. 650–656, proof of Theorem 1, p. 651; and for the latter case see my paper Concerning Menger Regular Curves, Fund. Math.12 (1928), Fundamental Accessibility Theorem.

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    G. T. Whyburn

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Presented to the American Mathematical Society, June 2, 1928.

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Whyburn, G.T. Concerning points of continuous curves defined by certain im kleinen properties. Math. Ann. 102, 313–336 (1930). https://doi.org/10.1007/BF01782349

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