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ISSN 1088-6826(e) ISSN 0002-9939(p)

Uncountable intersections of open sets under CPA $_{\mathrm{prism}}$

Author(s): Krzysztof Ciesielski; Janusz Pawlikowski
Journal: Proc. Amer. Math. Soc. 132 (2004), 3379-3385.
MSC (2000): Primary 03E35; Secondary 03E17
Posted: June 2, 2004
MathSciNet review: 2073315
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Abstract | References | Similar articles | Additional information

Abstract: We prove that the Covering Property Axiom CPA $_{\mathrm{prism}}$ , which holds in the iterated perfect set model, implies the following facts.

If is an intersection of $\omega_1$ -many open sets of a Polish space and has cardinality continuum, then contains a perfect set.
There exists a subset of the Cantor set which is an intersection of $\omega_1$ -many open sets but is not a union of $\omega_1$ -many closed sets.

The example from the second fact refutes a conjecture of Brendle, Larson, and Todorcevic.

References:

1.: J. Brendle, An e-mail to J. Pawlikowski, June 11, 2000.
2.: J. Brendle, P. Larson, S. Todorcevic, Rectangular axioms, perfect set properties and decomposition, preprint of November, 2002.
3.: K. Ciesielski, Set Theory for the Working Mathematician, London Math. Soc. Stud. Texts 39, Cambridge Univ. Press 1997. MR 99c:04001
4.: K. Ciesielski, J. Pawlikowski, Crowded and selective ultrafilters under the Covering Property Axiom, J. Appl. Anal. 9(1) (2003), 19-55. (Preprint available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html.)
5.: K. Ciesielski, J. Pawlikowski, Small coverings with smooth functions under the Covering Property Axiom, Canad. J. Math., to appear. (Preprint available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html.)
6.: K. Ciesielski, J. Pawlikowski, Covering Property Axiom CPA. A combinatorial core of the iterated perfect set model. To appear in Cambridge Tracts in Mathematics, Cambridge Univ. Press. (Preprint available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html.)
7.: K. Ciesielski, J. Pawlikowski, Covering Property Axiom CPA $_{\mathrm cube}$ and its consequences, Fund. Math. 176(1) (2003), 63-75. (Preprint available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html.) MR 2004b:03076
8.: V. Kanovei, Non-Glimm-Effros equivalence relations at second projective level, Fund. Math. 154 (1997), 1-35. MR 99j:03040
9.: A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, Berlin 1995. MR 96e:03057
10.: A. Nowik, Possibly there is no uniformly completely Ramsey null set of size $2^{\omega}$ , Colloq. Math. 93 (2002), 251-258. (Preprint available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html.) MR 2003j:03057
11.: J. Zapletal, Cardinal Invariants and Descriptive Set Theory, Mem. Amer. Math. Soc. 167 (2004).

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Additional Information:

Krzysztof Ciesielski
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
Email: K_Cies@math.wvu.edu

Janusz Pawlikowski
Affiliation: Department of Mathematics, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Email: pawlikow@math.uni.wroc.pl

DOI: 10.1090/S0002-9939-04-07475-1
PII: S 0002-9939(04)07475-1
Keywords: Uncountable intersections of open sets.
Received by editor(s): March 3, 2003
Received by editor(s) in revised form: July 27, 2003
Posted: June 2, 2004
Additional Notes: The work of the first author was partially supported by NATO Grant PST.CLG.977652 and by a 2002/03 West Virginia University Senate Research Grant.
Communicated by: Alan Dow
Copyright of article: Copyright 2004, American Mathematical Society

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