On open maps of Borel sets

A. Ostrovsky

Fundamenta Mathematicae (1995)

  • Volume: 146, Issue: 3, page 203-213
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Ostrovsky, A.. "On open maps of Borel sets." Fundamenta Mathematicae 146.3 (1995): 203-213. <http://eudml.org/doc/212062>.

@article{Ostrovsky1995,
abstract = {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.},
author = {Ostrovsky, A.},
journal = {Fundamenta Mathematicae},
keywords = {open maps; Borel sets; analytic sets; space of the first category; space of the second category; Baire space},
language = {eng},
number = {3},
pages = {203-213},
title = {On open maps of Borel sets},
url = {http://eudml.org/doc/212062},
volume = {146},
year = {1995},
}

TY - JOUR
AU - Ostrovsky, A.
TI - On open maps of Borel sets
JO - Fundamenta Mathematicae
PY - 1995
VL - 146
IS - 3
SP - 203
EP - 213
AB - 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.
LA - eng
KW - open maps; Borel sets; analytic sets; space of the first category; space of the second category; Baire space
UR - http://eudml.org/doc/212062
ER -

References

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  1. [1] F. van Engelen and J. van Mill, Borel sets in compact spaces: some Hurewicz-type theorems, Fund. Math. 124 (1984), 271-286. Zbl0559.54034
  2. [2] R. Engelking, General Topology, PWN, Warszawa, 1977. 
  3. [3] F. Hausdorff, Über innere Abbildungen, Fund. Math. 23 (1934), 279-291. Zbl60.0510.02
  4. [4] W. Hurewicz, Relativ perfekte Teile von Punktmengen und Mengen (A), ibid. 12 (1928), 78-109. 
  5. [5] L. V. Keldysh, On open maps of analytic sets, Dokl. Akad. Nauk SSSR 49 (1945), 646-648 (in Russian). 
  6. [6] K. Kuratowski, Topology, Vol. I, Academic Press, 1976. 
  7. [7] S. V. Medvedev, Zero-dimensional homogeneous Borel sets, Dokl. Akad. Nauk SSSR 283 (1985), 542-545 (in Russian). Zbl0604.54037
  8. [8] J. van Mill, Characterization of some zero-dimensional separable metric spaces, Trans. Amer. Math. Soc. 264 (1981), 205-215. Zbl0493.54018
  9. [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. [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. [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. [12] A. V. Ostrovsky, Cartesian product of F I I -spaces and analytic sets, Vestnik Moskov. Univ. Ser. Mat. 1975 (2), 29-34 (in Russian). 
  13. [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. [14] J. Saint Raymond, La structure borélienne d'Effros est-elle standard?, Fund. Math. 100 (1978), 201-210. Zbl0434.54028

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