Uncountable intersections of open sets under CPA$_{\mathrm {prism}}$
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- by Krzysztof Ciesielski and Janusz Pawlikowski PDF
- Proc. Amer. Math. Soc. 132 (2004), 3379-3385 Request permission
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.
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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
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3379-3385
- MSC (2000): Primary 03E35; Secondary 03E17
- DOI: https://doi.org/10.1090/S0002-9939-04-07475-1
- MathSciNet review: 2073315