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Concerning continuous curves in the plane

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References

  1. R. L. Moore, A theorem concerning continuous curves, Bull. Amer. Math. Soc., 2d. series,23 (Febr. 1917), S. 233–236. See also H. Tietze, Über stetige Kurven, Jordansche Kurvenbögen und geschlossene Jordansche Kurven, Math. Zeitschr.5 (1919), S. 284–291; and S. Mazurkiewicz, Sur les lignes de Jordan, Fundamenta Mathematicae1 (1920), S. 166–209. In this article, Mazurkiewicz establishes numerous results and indicates that some of them were published earlier in a journal (C. R. Soc. Sc. Varsovie) to which I do not at present have access.

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  2. N. J. Lennes, Curves in non-metrical analysis situs with an application in the calculus of variations, American Journal of Mathematics33, (1911), S. 287–326.

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  3. A point is said to be a limit point of a point-setM if every circle which enclosesP encloses at least one point ofM distinct fromP.

  4. Obviously every point-set which is connected in the strong sense is also connected in the weak sense and a closed point-set which is connected in the weak sense is also connected in the strong sense. It is accordingly allowable to speak of a “closed, connected point-set” without specifying in which sense the term connected is used.

  5. Cf Hans Hahn and S. Mazurkiewicz, loc. cit. Obviously every point-set which is connected in the strong sense is also connected in the weak sense and a closed point-set which is connected in the weak sense is also connected in the strong sense. It is accordingly allowable to speak of a “closed, connected point-set” without specifying in which sense the term connected is used.

  6. Transactions of the American Mathematical Society17 (1916), S. 131–164. This paper will be referred to as F. A.

  7. Hereafter in this paper a set of points will be said to be connected if it is connected in the weak sense.

  8. Cf. F. Hausdorff, Grundzüge der Mengenlehre, Veit and Co., Leipzig 1914, S. 344, VII.

  9. A. Schoenflies, Die Entwickelung der Lehre von den Punktmannigfaltigkeiten. Zweiter Teil, Leipzig 1908, S. 237.

  10. Über den Rand der einfach zusammenhängenden ebenen Gebiete. Math. Zeitschr.9 (1921), S. 64 (73).

  11. The point-set α-A O B certainly exists. For otherwise the simple continuous areA O B would be the complete boundary of a limited domain, which is clearly impossible.

  12. See my paper Concerning simple continuous curves, Transactions of the American Mathematical Society,21 (1920), S. 342.

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  13. Two pointsA andB are said to be separated from each other by the closed point-setM if every simple continuous arc fromA toB contains a point ofM distinct fromA and fromB.

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  1. Austin, Texas., USA

    Robert L. Moore

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  1. Robert L. Moore
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Moore, R.L. Concerning continuous curves in the plane. Math Z 15, 254–260 (1922). https://doi.org/10.1007/BF01494397

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