Uncountable intersections of open sets under CPA_{𝑝𝑟𝑖𝑠𝑚}

Ciesielski, Krzysztof; Pawlikowski, Janusz

doi:10.1090/S0002-9939-04-07475-1

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.

References

J. Brendle, An e-mail to J. Pawlikowski, June 11, 2000.
J. Brendle, P. Larson, S. Todorcevic, Rectangular axioms, perfect set properties and decomposition, preprint of November, 2002.
Krzysztof Ciesielski, Set theory for the working mathematician, London Mathematical Society Student Texts, vol. 39, Cambridge University Press, Cambridge, 1997. MR 1475462, DOI 10.1017/CBO9781139173131
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.)
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.)
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.)
Krzysztof Ciesielski and Janusz Pawlikowski, Covering property axiom $\rm CPA_{cube}$ and its consequences, Fund. Math. 176 (2003), no. 1, 63–75. MR 1903045, DOI 10.4064/fm176-1-5
Vladimir Kanovei, Non-Glimm-Effros equivalence relations at second projective level, Fund. Math. 154 (1997), no. 1, 1–35. MR 1472849, DOI 10.4064/fm-154-1-1-35
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Andrzej Nowik, Possibly there is no uniformly completely Ramsey null set of size $2^\omega$, Colloq. Math. 93 (2002), no. 2, 251–258. MR 1930802, DOI 10.4064/cm93-2-4
J. Zapletal, Cardinal Invariants and Descriptive Set Theory, Mem. Amer. Math. Soc. 167 (2004).