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On a problem in the theory of aggregates

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  • Law
  • Mathematics

Abstract

On a problem in the theory of aggregates COMPOSITIO MATHEMATICA D. LÁZÁR On a problem in the theory of aggregates Compositio Mathematica, tome 3 (1936), p. 304-304. <http://www.numdam.org/item?id=CM_1936__3__304_0> © Foundation Compositio Mathematica, 1936, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On a problem in the theory of aggregates by D. Lázár Szeged The problem dealt with in the following paper has been raised by P. Turàn. To every point of the interval 0 x 1 we adjoin a finite number of points of the same interval. This means, that we define a function y = 99(x) for o x 1 where o y 1, ~(x ) takes a finite number of values for every x, and the equation x = 99(x) is impossible. Two points x and y are called independent if neither of the two equations y = 99(x) and x = ~(y) holds. 1 am going to prove the following theorem. We can find a set of points in the interval 0 x 1 with the power of the continuum, so that any pair of its points is independent. The theorem, that there exist countably many points with the above property, has been proved by Mr. G. Grünwald 1) in a quite elementary way, using a theorem of Ramsay 2). To every point x we adjoin an interval containing x and having endpoints with rational coordinates, in such a manner that all values of ~(x ) are situated outside of this interval. We assert that there is an interval which is adjoined to a set of points of the power of the continuum. This assertion follows immediately from the fact that intervals with rational e

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