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NULL SETS AND COMBINATORIAL COVERING PROPERTIES

Part of: Set theory Fairly general properties

Published online by Cambridge University Press:  15 June 2021

PIOTR SZEWCZAK  and
TOMASZ WEISS
PIOTR SZEWCZAK
Affiliation:
INSTITUTE OF MATHEMATICS FACULTY OF MATHEMATICS AND NATURAL SCIENCE COLLEGE OF SCIENCES CARDINAL STEFAN WYSZYŃSKI UNIVERSITY IN WARSAW, WÓYCICKIEGO 1/3, 01–938 WARSAW, POLANDE-mail:p.szewczak@wp.plE-mail:tomaszweiss@o2.plURL: http://piotrszewczak.pl
TOMASZ WEISS
Affiliation:
INSTITUTE OF MATHEMATICS FACULTY OF MATHEMATICS AND NATURAL SCIENCE COLLEGE OF SCIENCES CARDINAL STEFAN WYSZYŃSKI UNIVERSITY IN WARSAW, WÓYCICKIEGO 1/3, 01–938 WARSAW, POLANDE-mail:p.szewczak@wp.plE-mail:tomaszweiss@o2.plURL: http://piotrszewczak.pl
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Abstract

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $ , a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.

Keywords

null setsnull-additive setsselection principlesγ-property

MSC classification

Primary: 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.)
Secondary: 03E35: Consistency and independence results 03E75: Applications of set theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 1231 - 1242
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Bartoszyński, T. and Judah, H., Set Theory: On the Structure of the Real Line, A. K. Peters, Wellesley, 1995.10.1201/9781439863466CrossRefGoogle Scholar
Bartoszyński, T. and Recław, I., Not every $\ \unicode{x3b3}$ -set is strongly meager . Contemporary Mathematics, vol. 192 (1996), pp. 25–29.CrossRefGoogle Scholar
Galvin, F. and Miller, A., γ-sets and other singular sets of real numbers . Topology and Its Applications, vol. 17 (1984), pp. 145–155.CrossRefGoogle Scholar
Gerlits, J. and Nagy, Z., Some properties of $C(X)$ , I . Topology and Its Applications, vol. 14 (1982), pp. 151–161.CrossRefGoogle Scholar
Just, W., Miller, A., Scheepers, M., and Szeptycki, P., The combinatorics of open covers II . Topology and Its Applications, vol. 73 (1996), pp. 241–266.CrossRefGoogle Scholar
Miller, A., Special subsets of the real line , Handbook of Set-Theoretic Topology (K. Kunen and J.E. Vaughan, editors), North-Holland, Amsterdam, 1984, pp. 201–233.10.1016/B978-0-444-86580-9.50008-2CrossRefGoogle Scholar
Orenshtein, T. and Tsaban, B., Linear $\ \sigma$ -additivity and some applications . Transactions of the American Mathematical Society, vol. 363 (2011), pp. 3621–3637.CrossRefGoogle Scholar
Osipov, A., Szewczak, P., and Tsaban, B., Strongly sequentially separable function spaces, via selection principles . Topology and Its Applications, vol. 270 (2020), Article no. 106942.CrossRefGoogle Scholar
Scheepers, M., Combinatorics of open covers I: Ramsey theory . Topology and Its Applications, vol. 69 (1996), pp. 31–62.CrossRefGoogle Scholar
Shelah, S., Every null-additive set is meager-additive . Israel Journal of Mathematics, vol. 89 (1995), pp. 357–376.CrossRefGoogle Scholar
Szewczak, P. and Tsaban, B., Products of Menger spaces: A combinatorial approach . Annals of Pure and Applied Logic, vol. 168 (2017), pp. 1–18.CrossRefGoogle Scholar
Szewczak, P. and Tsaban, B., Products of general Menger spaces . Topology and Its Applications, vol. 255 (2019), pp. 41–55.CrossRefGoogle Scholar
Szewczak, P. and Włudecka, M., Unbounded towers and products . Annals of Pure and Applied Logic, vol. 172 (2021), Article no. 102900.CrossRefGoogle Scholar
Tsaban, B., Strong $\ \unicode{x3b3}$ -sets and other singular spaces . Topology and Its Applications, vol. 153 (2005), pp. 620–639.10.1016/j.topol.2005.01.033CrossRefGoogle Scholar
Tsaban, B., Menger’s and Hurewicz’s problems: Solutions from “The Book” and refinements . Contemporary Mathematics, vol. 533 (2011), pp. 211–226.CrossRefGoogle Scholar
Weiss, T., Addendum to “On meager additive and null additive sets in the Cantor space $\ {2}^{\omega }$ and in $\ \mathbb{R}$ ”. Bulletin of the Polish Academy of Sciences. Mathematics, vol. 57 (2009), pp. 91–99.CrossRefGoogle Scholar
Zindulka, O., Strong measure zero and the like via selection principles . Proceedings of the Conference Frontiers of Selection Principles, Poland, 2017, Plenary Lecture. Slides available at http://selectionprinciples.com/Talks/Zindulka.pdf.Google Scholar