Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Kenilworth

Author(s): Tamás Mátrai
Journal: Proc. Amer. Math. Soc. 137 (2009), 1115-1125.
MSC (2000): Primary 03E15; Secondary 54H05, 28A05
Posted: October 23, 2008
MathSciNet review: 2457453
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Abstract: We construct a $G_{\delta}$ $\sigma$ -ideal $\mathcal{I}$ of compact subsets of $2^{\omega}$ such that $\mathcal{I}$ contains all the singletons but there is no dense $G_{\delta}$ set $D \subseteq 2^{\omega}$ such that $\{K \subseteq D \colon K\textrm{ compact}\} \subseteq \mathcal{I}$ . This answers a question of A. S. Kechris in the negative.

References:

1.: A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156 (Springer-Verlag, New York, 1995). MR 1321597 (96e:03057)
2.: A. S. Kechris, Hereditary properties of the class of closed sets of uniqueness for trigonometric series, Israel J. Math. 73 (1991), no. 2, 189-198. MR 1135211 (93c:42008)
3.: A. S. Kechris, The descriptive set theory of $\sigma$ -ideals of compact sets, Logic Colloq. '88 (Padova, 1988), 117-138, Stud. Logic Found. Math., 127, North-Holland, Amsterdam, 1989. MR 1015324 (90h:03032)
4.: A. S. Kechris, A. Louveau, W. H. Woodin, The structure of $\sigma$ -ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), no. 1, 263-288. MR 879573 (88f:03042)
5.: E. Matheron, M. Zelený, Descriptive set theory of families of small sets, Bull. Symb. Logic 13 (2007), no. 4, 482-537. MR 2369671
6.: S. Solecki, $G_\delta$ Ideals of Compact Sets, preprint.
7.: C. E. Uzcátegui A., The covering property for $\sigma$ -ideals of compact sets, Fund. Math. 141 (1992), no. 2, 119-146. MR 1183328 (94a:03077)

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Additional Information:

Tamás Mátrai
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda Street 13-15, H-1053 Budapest, Hungary
Address at time of publication: University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada
Email: matrait@renyi.hu

DOI: 10.1090/S0002-9939-08-09615-9
PII: S 0002-9939(08)09615-9
Keywords: $G_{\delta }$ $\sigma $-ideal of compact sets, singleton, ideal extension, covering property
Received by editor(s): November 14, 2007,
Received by editor(s) in revised form: March 9, 2008, and April 14, 2008
Posted: October 23, 2008
Additional Notes: This research was partially supported by the OTKA grants F 43620, K 49786, K 61600 and by the József Öveges Program of the National Office for Research and Technology.
Communicated by: Julia Knight
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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