Parametrized Cichoń's diagram and small sets

Fundamenta Mathematicae (1995)

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

top

Abstract

top
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 $e,f:{w}^{w}\to {w}^{w}$ such that  • N is ${G}_{\delta }$ and ${N}_{x}:x\in {w}^{w}$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of ${2}^{w}$;  • M is ${F}_{\sigma }$ and ${M}_{x}:x\in {w}^{w}$ is a basis for the ideal of meager subsets of ${2}^{w}$;  •$\forall x,y{N}_{e\left(x\right)}\subseteq {N}_{y}⇒{M}_{x}\subseteq {M}_{f\left(y\right)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. ${G}_{\delta }$) sets $B\subseteq X×{2}^{w}$ with all vertical sections null, ${\cup }_{x\in X}{B}_{x}$ is null, then for all Borel (resp. ${F}_{\sigma }$) sets $B\subseteq X×{2}^{w}$ with all vertical sections meager, ${\cup }_{x\in X}{B}_{x}$ is meager;  •if there exists a Borel (resp. a “nice” ${G}_{\delta }$) set $B\subseteq X×{2}^{w}$ such that ${B}_{x}:x\in X$ is a basis for measure zero sets, then there exists a Borel (resp. ${F}_{\sigma }$) set $B\subseteq X×{2}^{w}$ such that ${B}_{x}:x\in X$ is a basis for meager sets

How to cite

top

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

top
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

top

NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.