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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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

Authors: Krzysztof Ciesielski and Janusz Pawlikowski
Journal: Proc. Amer. Math. Soc. 132 (2004), 3379-3385
MSC (2000): Primary 03E35; Secondary 03E17
Published electronically: 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 $G$ is an intersection of $\omega_1$-many open sets of a Polish space and $G$ has cardinality continuum, then $G$ contains a perfect set.

  • There exists a subset $G$ 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 [Enhancements On Off] (What's this?)

  • 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. Krzysztof Ciesielski, Set theory for the working mathematician, London Mathematical Society Student Texts, vol. 39, Cambridge University Press, Cambridge, 1997. MR 1475462 (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:
  • 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:
  • 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:
  • 7. Krzysztof Ciesielski and Janusz Pawlikowski, Covering property axiom 𝐶𝑃𝐴_{𝑐𝑢𝑏𝑒} and its consequences, Fund. Math. 176 (2003), no. 1, 63–75. MR 1903045 (2004b:03076),
  • 8. Vladimir Kanovei, Non-Glimm-Effros equivalence relations at second projective level, Fund. Math. 154 (1997), no. 1, 1–35. MR 1472849 (99j:03040)
  • 9. Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • 10. Andrzej Nowik, Possibly there is no uniformly completely Ramsey null set of size 2^{𝜔}, Colloq. Math. 93 (2002), no. 2, 251–258. MR 1930802 (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

Janusz Pawlikowski
Affiliation: Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

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
Published electronically: 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
Article copyright: © Copyright 2004 American Mathematical Society

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