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Tomasz Weiss A note on the intersection ideal M ∩ N Comment.Math.Univ.Carolin. 54,3 (2013) 437 –445.

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Tomasz Weiss

A note on the intersection ideal M ∩ N

Comment.Math.Univ.Carolin. 54,3 (2013) 437 –445.

Abstract: We prove among other theorems that it is consistent with

ZF C

that there exists a set

X ⊆

2

ω

which is not meager additive, yet it satisfies the following property:

for each

Fσ

measure zero set

F

,

X

+

F

belongs to the intersection ideal

M ∩ N

.

Keywords:

Fσ

measure zero sets; intersection ideal

M ∩ N

; meager additive sets; sets perfectly meager in the transitive sense;

γ-sets

AMS Subject Classification: 03E05, 03E17 References

[1] Bartoszy´nski T., Judah H.,Set Theory, AK Peters, Wellesley, Massachusetts, 1995.

[2] Bartoszy´nski T., Rec law I.,Not everyγ-set is strongly meager, Contemp. Math., 192, Amer.

Math. Soc. Providence, RI, 1996, pp. 25–29.

[3] Bartoszy´nski T., Shelah S.,Strongly meager sets of size continuum, Arch. Math. Logic42 (2003), 769–779.

[4] Galvin F., Miller A., γ-sets and other singular sets of real numbers, Topology Appl.17 (1984), 145–155.

[5] Kraszewski J., Everywhere meagre and everywhere null sets, Houston J. Math.35(2009), no. 1, 103–111.

[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.

[7] Nowik A., Remarks about transitive version of perfectly meager sets, Real Anal. Exchange 22(1996/97), no. 1, 406–412.

[8] Nowik A., Scheepers M., Weiss T., The algebraic sum of sets of real numbers with strong measure zero sets, J. Symbolic Logic63(1998), 301–324.

[9] Nowik A., Weiss T.,Some remarks on totally imperfect sets, Proc. Amer. Math. Soc. 132 (2004), no. 1, 231–237.

[10] Pawlikowski J.,A characterization of strong measure zero sets, Israel J. Math.93(1996), 171–183.

[11] Pawlikowski J., Sabok M.,Two stars, Arch. Math. Logic47(2008), no. 7–8, 673–676.

[12] Zindulka O.,Small sets of reals through the prism of fractal dimensions, preprint, 2010.

[13] Cohen reals and strong measure zero sets– MathOverflow.15.

http://mathoverflow.net/questions/63497/ cohen-reals-and-strong-measure-zero-sets

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