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## Fundamenta Mathematicae

1995 | 147 | 2 | 135-155
Tytuł artykułu

### Parametrized Cichoń's diagram and small sets

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
135-155
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-03-24
poprawiono
1994-10-05
Twórcy
autor
• Department of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
autor
• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
Bibliografia
• [AR] A. Andryszczak and I. Recław, A note on strong measure zero sets, Acta Univ. Carolin. 34 (2) (1993), 7-9.
• [B1] T. Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc. 281 (1984), 209-213.
• [B2] T. Bartoszyński, Combinatorial aspects of measure and category, Fund. Math. 127 (1987), 225-239.
• [BJ] T. Bartoszyński and H. Judah, Measure and Category in Set Theory, a forthcoming book.
• [BR] T. Bartoszyński and I. Recław, Not every γ-set is strongly meager, preprint.
• [BSh] T. Bartoszyński and S. Shelah, Closed measure zero sets, Ann. Pure Appl. Logic 58 (1992), 93-110.
• [Bl] A. Blass, Questions and answers - a category arising in Linear Logic, Complexity Theory and Set Theory, preprint.
• [C] T. Carlson, Strong measure zero and strongly meager sets, Proc. Amer. Math. Soc. 118 (1993), 577-586.
• [F1] D. H. Fremlin, On the additivity and cofinality of Radon measures, Mathematika 31 (1984) (2) (1985), 323-335.
• [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).
• [FMi] D. H. Fremlin and A. Miller, On some properties of Hurewicz, Menger and Rothberger, Fund. Math. 129 (1988), 17-33.
• [G] F. Galvin, Indeterminacy of point-open games, Bull. Acad. Polon. Sci. 26 (1978), 445-449.
• [GMi] F. Galvin and A. W. Miller, γ-sets and other singular sets of real numbers, Topology Appl. 17 (1984), 145-155.
• [GMS] F. Galvin, J. Mycielski and R. M. Solovay, Strong measure zero sets, Notices Amer. Math. Soc. 26 (1979), A-280.
• [Ke] A. Kechris, Lectures on Classical Descriptive Set Theory, a forthcoming book.
• [L] R. Laver, On the consistency of Borel's conjecture, Acta Math. 137 (1976), 151-169.
• [Mi1] A. W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575-584.
• [Mi2] A. W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114.
• [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.
• [Mi4] A. W. Miller, On the length of Borel hierarchies, Ann. Math. Logic 16 (1979), 233-267.
• [P1] J. Pawlikowski, Lebesgue measurability implies Baire property, Bull. Sci. Math. (2) 109 (1985), 321-324.
• [P2] J. Pawlikowski, Every Sierpiński set is strongly meager, Arch. Math. Logic, to appear.
• [P3] J. Pawlikowski, Property C'', strongly meager sets and subsets of the plane, preprint.
• [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.
• [RaSt] J. Raisonnier and J. Stern, The strength of measurability hypothesis, ibid. 50 (1985), 337-349.
• [R1] I. Recław, Every Lusin set is undetermined in the point-open game, Fund. Math. 144 (1994), 43-54.
• [R2] I. Recław, Cichoń's diagram and continuum hypothesis, circulated manuscript, 1992.
• [R3] I. Recław, On small sets in the sense of measure and category, Fund. Math. 133 (1989), 254-260.
• [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.
Typ dokumentu
Bibliografia
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