# On open maps of Borel sets

Fundamenta Mathematicae (1995)

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

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topOstrovsky, 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

top- [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] R. Engelking, General Topology, PWN, Warszawa, 1977.
- [3] F. Hausdorff, Über innere Abbildungen, Fund. Math. 23 (1934), 279-291. Zbl60.0510.02
- [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). Zbl0604.54037
- [8] J. van Mill, Characterization of some zero-dimensional separable metric spaces, Trans. Amer. Math. Soc. 264 (1981), 205-215. Zbl0493.54018
- [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 ${F}_{I}I$-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. Zbl0434.54028

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