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Concerning upper semi-continuous collections. (English) JFM 55.0317.04

In einer früheren Arbeit (Transactions A. M. S. 27 (1925), 416-428; F. d. M. 51, 464 (JFM 51.0464.*)-465) hat Verf. den Begriff der “upper semi-continuous collection” von Kontinuen eingeführt und bewiesen, daß eine solche Menge von paarweise fremden kompakten Kontinuen (Entartung auf einen Punkt ist zulässig), die eine Kugeloberfläche ausfüllen und von denen keines die Kugel zerlegt, bei geeigneter Definition eines Limesbegriffes, der Kugel homöomorph ist.
In vorliegender Arbeit läßt Verf. die Voraussetzung, daß keines der Kontinua, der upper semi-continuous collection die Kugel zerlegen soll, fallen und gelangt zu folgendem Resultat: Ist \(G\) eine upper semi-continuous collection von paarweise fremden Kontinuen, die eine Kugel \(S\) erfüllen, so ist der Raum \(S_G\) der Elemente von \(G\) homöomorph einem “cactoid”. Dabei ist ein “cactoid” oder “opuntioid” eine stetige Kurve \(M\) im dreidimensionalen Raum, deren nicht in einen Punkt entartete maximale zyklische Teilmengen (d. h. maximale zyklische Kurven oder Punkte, die keiner maximalen zyklischen Kurve angehören) Flächen vom Zusammenhang der Kugel sind, und die überdies die Eigenschaft hat, daß kein Punkt von \(M\) im Innern eines beschränkten Komplementärgebietes irgendeines Teilkontinuums von \(M\) liegt. Ein cactoid \(M\) heißt “aspiculate”, wenn es keinen Bogen \(t\) in \(M\) gibt von der Eigenschaft, daß \(t\) bis auf die Endpunkte frei ist von Häufungspunkten von \(M- t\). Ein aspiculate cactoid ist einfach, wenn keine zwei Zerschneidungspunkte von \(M\) in \(M\) durch unendlich viele Punkte voneinander getrennt werden.
Es gilt weiter: Ist \(G\) eine upper-semi-continuous collection von paarweise fremden Kontinuen, die die Kugel \(S\) erfüllen, und besteht \(G\) mit Ausnahme einer Folge von Kontinuen mit gegen Null abnehmenden Durchmessern nur aus Punkten, so ist der Raum \(S_G\) homöomorph einem einfachen aspiculate cactoid.
Von diesen beiden Sätzen gelten auch gewisse Umkehrungen: Zu einem cactoid und einer Kugel \(S\) gibt es immer eine upper semi-continuous collection von Kontinuen auf \(S\), so daß der Raum, dessen Elemente die Kontinua von \(G\) sind, dem cactoid \(K\) homöomorph ist, und daß alle nicht entarteten Kontinua von \(G\) die Kugel \(S\) zerlegen. – Zu einem einfachen aspiculate cactoid \(K\) gibt es eine upper semi-continuous collection \(G\) von paarweise fremden Kontinuen auf \(S\) so, daß der Raum, dessen Elemente die Kontinua von \(G\) sind, zu \(K\) homöomorph ist und überdies alle Elemente von \(G\) bis auf eine Folge von Kontinuen mit gegen Null abnehmenden Durchmessern zu Punkten entartet sind.

Citations:

JFM 51.0464.*
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[1] Concerning upper semi-continuous collections of continua, Transactions of the American Mathematical Society, vol. 27 (1925), pp. 416–428.
[2] A cyclic subset of a continuous curveM is a subsetK ofM such that every two points ofK belong to some simple closed curve which is a subset ofK and a cyclic subset ofM is said to be maximal if it is not a proper subset of any other cyclic subset ofM. See the following papers by G. T. Whyburn: Cyclically connected continuous curves. Proceedings of the National Academy of Sciences, vol. 13 (1927), pp. 31–38; Some properties of continuous curves, Bulletin of the American Mathematical Society, vol. 33 (1927), pp. 305–308; Concerning the structure of a continuous curve, American Journal of Mathematics, vol. L (1928), pp. 167–194. The term maximal cyclic subset as used here is more inclusive than Whyburns maximal cyclic continuous curve only in that a point may be a maximal cyclic subset ofM but no point is a cyclic continuous curve in the sense of Whyburn. On the other hand Whyburn’s cyclic element is more inclusive than maximal cyclic subset in that a cut point ofM that belongs to a maximal cyclic continuous curve ofM is a cyclic element ofM but not a maximal cyclic subset ofM. This slight departure from Whyburn’s terminology is made, in the present paper, largely for contextual reasons. A maximal cyclic subset ofM will be said to be degenerate if it consists of a single point. · doi:10.1073/pnas.13.2.31
[3] Vietoris has shown that if the spaceS G is one dimensional then it is an acyclic continuous curve. Every such curve is a cactoid whose maximal cyclic subsets are all points. Cf. L. Vietoris, Über stetige Abbildungen einer Kugelfläche, Proceedings of the Royal Academy of Sciences of Amsterdam, vol. 29 (1926), pp. 443–453.
[4] R. L. Moore, loc. cit. (1923), pp. 101–106, Theorem 26. While this theorem was established for a Euclidean spaceS ofn-dimensions, it is clear that (if the last definition of Page 417 is omitted) the same argument suffices to establish it for the case where the spaceS is a spherical surface or, indeed, any metric space which itself satisfies Axioms 1, 2, 4 and 51 and Theorem 4 of my Foundations article. That every space satisfying Axiom 1 and Theorem 4 is necessarily metric may be easily seen with the help of the Urysohn-Tychonoff theorem to the effect that every regular and perfectly separable Hausdorff space is metric. See P. Urysohn, Mathematische Annalen, vol. 94 (1925), pp. 309–315 and A. Tychonoff, Ibid. vol. 95 (1926), pp. 139–142. It has been shown by Chittenden that in order that a topological space in which there are no isolated points should be metric and separable it is necessary and sufficient that is should satisfy my Axiom 1. Cf. E. W. Chittenden, On the metrization problem and related problems in the theory of abstract sets, Bulletin of the American Mathematical Society, vol. 33 (1927), pp. 13–34. · doi:10.1073/pnas.9.4.101
[5] Transactions of the American Mathematical Society, vol. 17 (1916), pp. 131–164.
[6] See S. Mazurkiewicz, Fundamenta Mathematicae, vol. 2 (1921). pp. 119–130 and R. L. Moore, Proceedings of the National Academy of Sciences, vol. 9 (1923), pp. 101–106.
[7] Cf. L. Vietoris,loc. cit., page 7, fourth paragraph.
[8] See T. Wazewski, Sur les courbes de Jordan ne renfermant aucune courbe simple fermée de Jordan, Annales de la Société Polonaise de Mathématique, vol. 2 (1923), pp. 49–170, and a later article, by H. M. Gehman, Transactions of the American Mathematical Society, vol. 29 (1927), pp. 553 bis 568.
[9] A sequence of point sets is said to be contracting if, for each positive numbere, all buta finite number of the point sets of that sequence are of diameter less thane.
[10] A simple chain of maximal cyclic subsets of a cactoidK is a collectiont of maximal cyclic subsets ofK such that ifM denotes the sum of all the elements oft thenM is a continuum containing two elementsA andB oft such thatA andB do not both belong to any proper connected subset ofM which is the sum of the elements of some collection of maximal cyclic subsets ofK. Cf. G. T. Whyburn, Concerning the structure, of a continuous curve, loc. cit., page 169.
[11] That the set of all such points is countable may be established by an argument similar to arguments employed to show that no acyclic continuous curve has more than a countable number of branch points. Cf. T. Wazewski, loc. cit., and K. Menger, Über reguläre Baumkurven, Mathematische Annalen, vol. 96 (1926), pp. 572–582.
[12] For eachX, all but possibly one of the continua ofU (X) are degenerate.
[13] See S. Mazurkiewicz, loc. cit., and R. L. Moore, loc. cit. Proceedings of the National Academy of Sciences, vol. 9 (1923), pp. 101–106.
[14] For the case whereK is an acyclic continuous curve see L. Vietoris loc. cit..
[15] The truth of this statement may be established by an argument having much in common with the argument given by R. L. Wilder to prove Theorem 15 of his paper Concerning continuous curves, Fundamenta Mathematica, vol 7 (1925), pp. 340–377.
[16] J. R. Kline, Concerning the sum of a countable infinity of mutually exclusive continua, Mathematische Zeitschrift, vol. 26 (1927), pp. 687–690. · doi:10.1007/BF01475482
[17] , Section IV, pages 427. and 428. · doi:10.1007/BF01475482
[18] Cf. Hausdorff, Grundzüge der Mengenlehre, page 248.
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