Uncountable intersections of open sets under CPA

Authors:
Krzysztof Ciesielski and Janusz Pawlikowski

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

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 , which holds in the iterated perfect set model, implies the following facts.

- If is an intersection of -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 -many open sets but is not a union of -many closed sets.

**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****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.**Krzysztof Ciesielski and Janusz Pawlikowski,*Covering property axiom 𝐶𝑃𝐴_{𝑐𝑢𝑏𝑒} and its consequences*, Fund. Math.**176**(2003), no. 1, 63–75. MR**1903045**, https://doi.org/10.4064/fm176-1-5**8.**Vladimir Kanovei,*Non-Glimm-Effros equivalence relations at second projective level*, Fund. Math.**154**(1997), no. 1, 1–35. MR**1472849****9.**Alexander S. Kechris,*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR**1321597****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**, https://doi.org/10.4064/cm93-2-4**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 Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Email:
pawlikow@math.uni.wroc.pl

DOI:
https://doi.org/10.1090/S0002-9939-04-07475-1

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