More remarks on the intersection ideal \(\mathcal{M}\cap\mathcal{N}\). (English) Zbl 1424.03020

Weiss, Tomasz

Commentat. Math. Univ. Carol. 59, No. 3, 311-316 (2018).

Given two ideal \(I\) and \(J\) on the Cantor set \(2^\omega\) we say that \(X\in(I,J)^*\) if for all \(A\) in \(I\) the sum \(A+X\) is in \(J\); also, \(I^*\) abbreviates \((I,I)^*\). Answering a question from [the author, Commentat. Math. Univ. Carol. 54, No. 3, 437–445 (2013; Zbl 1289.03031)], the author shows that \((\mathcal{M}\cap \mathcal{N})^*\subseteq \mathcal{N}^*\), where \(\mathcal{M}\) and \(\mathcal{N}\) are the ideals of meager and null sets, respectively. In addition, there is a provisional solution to the question whether it is consistent that every member of \((\mathcal{E},\mathcal{M})^*\) is countable, where \(\mathcal{E}\) is the \(\sigma\)-ideal generated by the \(F_\sigma\)-sets of measure zero.

Reviewer: K. P. Hart (Delft)

MSC:

03E05	Other combinatorial set theory
03E17	Cardinal characteristics of the continuum
03E15	Descriptive set theory

Keywords:

meager set; null set; additive meager set; additive null set

Citations:

Zbl 1289.03031

References:

[1]	Bartoszyński T., Remarks on small sets of reals, Proc. Amer. Math. Soc 131 (2003), no. 2, 625-630 · Zbl 1017.03027
[2]	Bartoszyński T.; Judah H., Set Theory. On the Structure of the Real Line, A K Peters, Wellesley, 1995 · Zbl 0834.04001
[3]	Goldstern M.; Kellner J.; Shelah S.; Wohofsky W., Borel conjecture and dual Borel conjecture, Trans. Amer. Math. Soc. 366 (2014), no. 1, 245-307 · Zbl 1298.03106
[4]	Orenshtein T.; Tsaban B., Linear \(σ\)-additivity and some applications, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3621-3637 · Zbl 1223.54032
[5]	Pawlikowski J., A characterization of strong measure zero sets, Israel J. Math. 93 (1996), 171-183 · Zbl 0857.28001
[6]	Weiss T., A note on the intersection ideal \({\mathcal{M}}∩ {\mathcal{N}}\), Comment. Math. Univ. Carolin. 54 (2013), no. 3, 437-445