Parametrized Cichoń's diagram and small sets

Janusz Pawlikowski; Ireneusz Recław

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 2, page 135-155
  • ISSN: 0016-2736

Abstract

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We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of w w × 2 w and continuous functions e , f : w w → w w such that  • N is G δ and N x : x ∈ w w , the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of 2 w ;  • M is F σ and M x : x ∈ w w is a basis for the ideal of meager subsets of 2 w ;  • ∀ x , y N e ( x ) ⊆ N y ⇒ M x ⊆ M f ( y ) . From this we derive that for a separable metric space X,  •if for all Borel (resp. G δ ) sets B ⊆ X × 2 w with all vertical sections null, ∪ x ∈ X B x is null, then for all Borel (resp. F σ ) sets B ⊆ X × 2 w with all vertical sections meager, ∪ x ∈ X B x is meager;  •if there exists a Borel (resp. a “nice” G δ ) set B ⊆ X × 2 w such that B x : x ∈ X is a basis for measure zero sets, then there exists a Borel (resp. F σ ) set B ⊆ X × 2 w such that B x : x ∈ X is a basis for meager sets

How to cite

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Pawlikowski, Janusz, and Recław, Ireneusz. "Parametrized Cichoń's diagram and small sets." Fundamenta Mathematicae 147.2 (1995): 135-155. <http://eudml.org/doc/212079>.

@article{Pawlikowski1995,
abstract = {We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w×2^w$ and continuous functions $e, f:w^w → w^w$ such that  • N is $G_δ$ and $\{N_x:x ∈ w^w\}$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_σ$ and $\{M_x:x ∈ w^w\}$ is a basis for the ideal of meager subsets of $2^w$;  •$∀x,y N_\{e(x)\} ⊆ N_y ⇒ M_x ⊆ M_\{f(y)\}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_δ$) sets $B ⊆ X×2^w$ with all vertical sections null, $∪_\{x ∈ X\}B_x$ is null, then for all Borel (resp. $F_σ$) sets $B ⊆ X×2^w$ with all vertical sections meager, $∪_\{x ∈ X\}B_x$ is meager;  •if there exists a Borel (resp. a “nice” $G_δ$) set $B ⊆ X×2^w$ such that $\{B_x:x ∈ X\}$ is a basis for measure zero sets, then there exists a Borel (resp. $F_σ$) set $B ⊆ X×2^w$ such that $\{B_x:x ∈ X\}$ is a basis for meager sets},
author = {Pawlikowski, Janusz, Recław, Ireneusz},
journal = {Fundamenta Mathematicae},
keywords = {measure zero set; Cichoń’s diagram; classes of small sets of reals; meager set},
language = {eng},
number = {2},
pages = {135-155},
title = {Parametrized Cichoń's diagram and small sets},
url = {http://eudml.org/doc/212079},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Pawlikowski, Janusz
AU - Recław, Ireneusz
TI - Parametrized Cichoń's diagram and small sets
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 2
SP - 135
EP - 155
AB - We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w×2^w$ and continuous functions $e, f:w^w → w^w$ such that  • N is $G_δ$ and ${N_x:x ∈ w^w}$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_σ$ and ${M_x:x ∈ w^w}$ is a basis for the ideal of meager subsets of $2^w$;  •$∀x,y N_{e(x)} ⊆ N_y ⇒ M_x ⊆ M_{f(y)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_δ$) sets $B ⊆ X×2^w$ with all vertical sections null, $∪_{x ∈ X}B_x$ is null, then for all Borel (resp. $F_σ$) sets $B ⊆ X×2^w$ with all vertical sections meager, $∪_{x ∈ X}B_x$ is meager;  •if there exists a Borel (resp. a “nice” $G_δ$) set $B ⊆ X×2^w$ such that ${B_x:x ∈ X}$ is a basis for measure zero sets, then there exists a Borel (resp. $F_σ$) set $B ⊆ X×2^w$ such that ${B_x:x ∈ X}$ is a basis for meager sets
LA - eng
KW - measure zero set; Cichoń’s diagram; classes of small sets of reals; meager set
UR - http://eudml.org/doc/212079
ER -

References

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  1. [AR] A. Andryszczak and I. Recław, A note on strong measure zero sets, Acta Univ. Carolin. 34 (2) (1993), 7-9. 
  2. [B1] T. Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc. 281 (1984), 209-213. Zbl0538.03042
  3. [B2] T. Bartoszyński, Combinatorial aspects of measure and category, Fund. Math. 127 (1987), 225-239. Zbl0635.04001
  4. [BJ] T. Bartoszyński and H. Judah, Measure and Category in Set Theory, a forthcoming book. 
  5. [BR] T. Bartoszyński and I. Recław, Not every γ-set is strongly meager, preprint. Zbl0838.03037
  6. [BSh] T. Bartoszyński and S. Shelah, Closed measure zero sets, Ann. Pure Appl. Logic 58 (1992), 93-110. Zbl0764.03018
  7. [Bl] A. Blass, Questions and answers - a category arising in Linear Logic, Complexity Theory and Set Theory, preprint. 
  8. [C] T. Carlson, Strong measure zero and strongly meager sets, Proc. Amer. Math. Soc. 118 (1993), 577-586. Zbl0787.03037
  9. [F1] D. H. Fremlin, On the additivity and cofinality of Radon measures, Mathematika 31 (1984) (2) (1985), 323-335. Zbl0551.28015
  10. [F2] D. H. Fremlin, Cichoń's diagram, in: Sém. d'Initiation à l'Analyse, G. Choquet, M. Rogalski and J. Saint-Raymond (eds.), Publ. Math. Univ. Pierre et Marie Curie, 1983/84, (5-01)-(5-13). 
  11. [FMi] D. H. Fremlin and A. Miller, On some properties of Hurewicz, Menger and Rothberger, Fund. Math. 129 (1988), 17-33. Zbl0665.54026
  12. [G] F. Galvin, Indeterminacy of point-open games, Bull. Acad. Polon. Sci. 26 (1978), 445-449. Zbl0392.90101
  13. [GMi] F. Galvin and A. W. Miller, γ-sets and other singular sets of real numbers, Topology Appl. 17 (1984), 145-155. 
  14. [GMS] F. Galvin, J. Mycielski and R. M. Solovay, Strong measure zero sets, Notices Amer. Math. Soc. 26 (1979), A-280. 
  15. [Ke] A. Kechris, Lectures on Classical Descriptive Set Theory, a forthcoming book. 
  16. [L] R. Laver, On the consistency of Borel's conjecture, Acta Math. 137 (1976), 151-169. Zbl0357.28003
  17. [Mi1] A. W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575-584. Zbl0527.03031
  18. [Mi2] A. W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114. 
  19. [Mi3] 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. 
  20. [Mi4] A. W. Miller, On the length of Borel hierarchies, Ann. Math. Logic 16 (1979), 233-267. Zbl0415.03038
  21. [P1] J. Pawlikowski, Lebesgue measurability implies Baire property, Bull. Sci. Math. (2) 109 (1985), 321-324. Zbl0593.03026
  22. [P2] J. Pawlikowski, Every Sierpiński set is strongly meager, Arch. Math. Logic, to appear. Zbl0871.04003
  23. [P3] J. Pawlikowski, Property C'', strongly meager sets and subsets of the plane, preprint. 
  24. [Ra] J. Raisonnier, A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel J. Math. 48 (1984), 48-56. Zbl0596.03056
  25. [RaSt] J. Raisonnier and J. Stern, The strength of measurability hypothesis, ibid. 50 (1985), 337-349. Zbl0602.03012
  26. [R1] I. Recław, Every Lusin set is undetermined in the point-open game, Fund. Math. 144 (1994), 43-54. Zbl0809.04002
  27. [R2] I. Recław, Cichoń's diagram and continuum hypothesis, circulated manuscript, 1992. 
  28. [R3] I. Recław, On small sets in the sense of measure and category, Fund. Math. 133 (1989), 254-260. Zbl0707.28001
  29. [V] P. Vojtáš, Generalized Galois-Tukey connections between explicit relations on classical objects of real analysis, in: Set Theory of the Reals, H. Judah (ed.), Israel Math. Conf. Proc. 6, 1993, 619-643. Zbl0829.03027

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