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1994-1995 | 146 | 3 | 203-213
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On open maps of Borel sets

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EN
We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers ℙ not $G_δ · F_σ$ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ ℙ of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. The answer to the question as stated is negative if X is Baire but Y is not Baire.
Twórcy
  • Marine Technical University, Lotzmanskaya Str. 3, St. Petersburg, 190006, Russia
Bibliografia
  • [1] F. van Engelen and J. van Mill, Borel sets in compact spaces: some Hurewicz-type theorems, Fund. Math. 124 (1984), 271-286.
  • [2] R. Engelking, General Topology, PWN, Warszawa, 1977.
  • [3] F. Hausdorff, Über innere Abbildungen, Fund. Math. 23 (1934), 279-291.
  • [4] W. Hurewicz, Relativ perfekte Teile von Punktmengen und Mengen (A), ibid. 12 (1928), 78-109.
  • [5] L. V. Keldysh, On open maps of analytic sets, Dokl. Akad. Nauk SSSR 49 (1945), 646-648 (in Russian).
  • [6] K. Kuratowski, Topology, Vol. I, Academic Press, 1976.
  • [7] S. V. Medvedev, Zero-dimensional homogeneous Borel sets, Dokl. Akad. Nauk SSSR 283 (1985), 542-545 (in Russian).
  • [8] J. van Mill, Characterization of some zero-dimensional separable metric spaces, Trans. Amer. Math. Soc. 264 (1981), 205-215.
  • [9] A. V. Ostrovsky, Concerning the Keldysh question about the structure of Borel sets, Mat. Sb. 131 (1986), 323-346 (in Russian); English transl.: Math. USSR-Sb. 59 (1988), 317-337.
  • [10] A. V. Ostrovsky, On open mappings of zero-dimensional spaces, Dokl. Akad. Nauk SSSR 228 (1976), 34-37 (in Russian); English transl.: Soviet Math. Dokl. 17 (1976), 647-654.
  • [11] A. V. Ostrovsky, On nonseparable τ-analytic sets and their mappings, Dokl. Akad. Nauk SSSR 226 (1976), 269-272 (in Russian); English transl.: Soviet Math. Dokl. 17 (1976), 99-102.
  • [12] A. V. Ostrovsky, Cartesian product of $F_II$-spaces and analytic sets, Vestnik Moskov. Univ. Ser. Mat. 1975 (2), 29-34 (in Russian).
  • [13] A. V. Ostrovsky, Continuous images of the product ℂ × ℚ of the Cantor perfect set ℂ and the rational numbers ℚ, in: Seminar on General Topology, Moskov. Gos. Univ., Moscow, 1981, 78-85 (in Russian).
  • [14] J. Saint Raymond, La structure borélienne d'Effros est-elle standard?, Fund. Math. 100 (1978), 201-210.
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bwmeta1.element.bwnjournal-article-fmv146i3p203bwm
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