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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?)

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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 Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: pawlikow@math.uni.wroc.pl

DOI: http://dx.doi.org/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
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