Previous |  Up |  Next

Article

MSC: 03E05, 03E17
Full entry | PDF   (0.2 MB)
Keywords:
$F_\sigma$ measure zero sets; intersection ideal $\mathcal M\cap \mathcal N$; meager additive sets; sets perfectly meager in the transitive sense; $\gamma$-sets
Summary:
We prove among other theorems that it is consistent with $ZFC$ that there exists a set $X\subseteq 2^\omega$ which is not meager additive, yet it satisfies the following property: for each $F_\sigma$ measure zero set $F$, $X+F$ belongs to the intersection ideal $\mathcal M\cap \mathcal N$.
References:
[1] Bartoszyński T., Judah H.: Set Theory. AK Peters, Wellesley, Massachusetts, 1995. MR 1350295
[2] Bartoszyński T., Recław I.: Not every $\gamma$-set is strongly meager. Contemp. Math., 192, Amer. Math. Soc. Providence, RI, 1996, pp. 25–29. DOI 10.1090/conm/192/02346 | MR 1367132 | Zbl 0838.03037
[3] Bartoszyński T., Shelah S.: Strongly meager sets of size continuum. Arch. Math. Logic 42 (2003), 769–779. DOI 10.1007/s00153-003-0184-0 | MR 2020043 | Zbl 1041.03034
[4] Galvin F., Miller A.: $\gamma$-sets and other singular sets of real numbers. Topology Appl. 17 (1984), 145–155. DOI 10.1016/0166-8641(84)90038-5 | MR 0738943 | Zbl 0551.54001
[5] Kraszewski J.: Everywhere meagre and everywhere null sets. Houston J. Math. 35 (2009), no. 1, 103–111. MR 2491870 | Zbl 1160.03028
[6] Miller A.: Special subsets of the real line. in Handbook of Set-Theoretic Topology, edited by K. Kunen and J.E. Vaughan, North-Holland, 1984, pp. 201–233. MR 0776624 | Zbl 0588.54035
[7] Nowik A.: Remarks about transitive version of perfectly meager sets. Real Anal. Exchange 22 (1996/97), no. 1, 406–412. MR 1433627
[8] Nowik A., Scheepers M., Weiss T.: The algebraic sum of sets of real numbers with strong measure zero sets. J. Symbolic Logic 63 (1998), 301–324. DOI 10.2307/2586602 | MR 1610427 | Zbl 0901.03036
[9] Nowik A., Weiss T.: Some remarks on totally imperfect sets. Proc. Amer. Math. Soc. 132 (2004), no. 1, 231–237. DOI 10.1090/S0002-9939-03-06997-1 | MR 2021267 | Zbl 1041.03035
[10] Pawlikowski J.: A characterization of strong measure zero sets. Israel J. Math. 93 (1996), 171–183. DOI 10.1007/BF02761100 | MR 1380640 | Zbl 0857.28001
[11] Pawlikowski J., Sabok M.: Two stars. Arch. Math. Logic 47 (2008), no. 7–8, 673–676. DOI 10.1007/s00153-008-0095-1 | MR 2448952 | Zbl 1152.28003
[12] Zindulka O.: Small sets of reals through the prism of fractal dimensions. preprint, 2010.
[13] HASH(0x9e03828): {\it Cohen reals and strong measure zero sets} – MathOverflow.15.\ http://mathoverflow.net/questions/63497/ cohen-reals-and-strong-measure-zero-sets.

Partner of