Strongly meager sets and subsets of the plane

Janusz Pawlikowski

Fundamenta Mathematicae (1998)

  • Volume: 156, Issue: 3, page 279-287
  • ISSN: 0016-2736

Abstract

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Let X ⊆ 2 w . Consider the class of all Borel F ⊆ X × 2 w with null vertical sections F x , x ∈ X. We show that if for all such F and all null Z ⊆ X, ∪ x ∈ Z F x is null, then for all such F, ∪ x ∈ X F x ≠ 2 w . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

How to cite

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Pawlikowski, Janusz. "Strongly meager sets and subsets of the plane." Fundamenta Mathematicae 156.3 (1998): 279-287. <http://eudml.org/doc/212273>.

@article{Pawlikowski1998,
abstract = {Let $X ⊆ 2^w$. Consider the class of all Borel $F ⊆ X×2^w$ with null vertical sections $F_x$, x ∈ X. We show that if for all such F and all null Z ⊆ X, $∪_\{x ∈ Z\}F_x$ is null, then for all such F, $∪_\{x ∈ X\}F_x≠2^w$. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].},
author = {Pawlikowski, Janusz},
journal = {Fundamenta Mathematicae},
keywords = {null set; Borel set; strongly meager set},
language = {eng},
number = {3},
pages = {279-287},
title = {Strongly meager sets and subsets of the plane},
url = {http://eudml.org/doc/212273},
volume = {156},
year = {1998},
}

TY - JOUR
AU - Pawlikowski, Janusz
TI - Strongly meager sets and subsets of the plane
JO - Fundamenta Mathematicae
PY - 1998
VL - 156
IS - 3
SP - 279
EP - 287
AB - Let $X ⊆ 2^w$. Consider the class of all Borel $F ⊆ X×2^w$ with null vertical sections $F_x$, x ∈ X. We show that if for all such F and all null Z ⊆ X, $∪_{x ∈ Z}F_x$ is null, then for all such F, $∪_{x ∈ X}F_x≠2^w$. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].
LA - eng
KW - null set; Borel set; strongly meager set
UR - http://eudml.org/doc/212273
ER -

References

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  1. [B] T. Bartoszyński, On covering the real line with null sets, Pacific J. Math. 131 (1988), 1-12. Zbl0643.03034
  2. [BJ] T. Bartoszyński and H. Judah, Borel images of sets of reals, Real Anal. Exchange 20 (1995), 1-23. 
  3. [BJ1] T. Bartoszyński and H. Judah, Set Theory: on the Structure of the Real Line, A. K. Peters, Wellesley, Mass., 1995. 
  4. [C] T. Carlson, Strong measure zero and strongly meager sets, Proc. Amer. Math. Soc. 118 (1993), 577-586. Zbl0787.03037
  5. [FM] D. H. Fremlin and A. Miller, On some properties of Hurewicz, Menger and Rothberger, Fund. Math. 129 (1988), 17-33. Zbl0665.54026
  6. [K] A. Kechris, Classical Descriptive Set Theory, Springer, 1995. 
  7. [M] A. W. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), Elsevier, 1984, 203-233. 
  8. [M1] A. W. Miller, Additivity of measure implies dominating reals, Proc. Amer. Math. Soc. 91 (1984), 111-117. Zbl0586.03042
  9. [P] J. Pawlikowski, Every Sierpiński set is strongly meager, Arch. Math. Logic 35 (1996), 281-285. Zbl0871.04003
  10. [PR] J. Pawlikowski and I. Recław, Parametrized Cichoń's diagram and small sets, Fund. Math. 147 (1995), 135-155. Zbl0847.04004

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