Bartoszynski, T. and S. Shelah, Closed measure zero sets, Annals of Pure and Applied Logic 58 93–110. We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of (...) the ideal of meager sets. (shrink)

We prove that the cofinality of the smallest covering of R by meager sets is bigger than the additivity of measure.

We conclude the discussion of additivity, Baire number, uniformity, and covering for measure and category by constructing the remaining 5 models. Thus we complete the analysis of Cichon's diagram.

We prove that the following are consistent with ZFC. 1. 2 ω = ℵ ω 1 + K C = ℵ ω 1 + K B = K U = ω 2 (for measure and category simultaneously). 2. 2 ω = ℵ ω 1 = K C (L) + K C (M) = ω 2 . This concludes the discussion about the cofinality of K C.

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions.

A set X⊆ℝ is strongly meager if for every measure zero set H, X+H ≠ℝ. Let [Formula: see text] denote the collection of strongly meager sets. We show that assuming [Formula: see text], [Formula: see text] is not an ideal.

We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.

We construct a model of ZFC satisfying the Dual Borel Conjecture in which there is a set of size ℵ₁ that does not have measure zero.

We show that it is consistent with ZFC that the intersection of some family of less than ultrafilters have measure zero. This answers a question of D. Fremlin.

Abstract.We will construct several models where there are no strongly meager sets of size 2ℵ0.

In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID=""ID="" Mathematics Subject Classification (2000): 03e35 RID=""ID="" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658.

We will construct several models where there are no strongly meager sets of size 2ℵ0.

We show that it is consistent with MA + ¬CH that the Forcing Axiom fails for all forcing notions in the class of ωω–bounding forcing notions with norms of [17].

Reviewed Works:Tomek Bartoszynski, Marion Scheepers, Set Theory, Annual Boise Extravaganza in Set Theory Conference, March 13-15, 1992, April 10-11, 1993, March 25-27, 1994, Boise State University, Boise, Idaho.R. Aharoni, A. Hajnal, E. C. Milner, Interval Covers of a Linearly Ordered Set.Eyal Amir, Haim Judah, Souslin Absoluteness, Uniformization and Regularity Properties of Projective Sets.Tomek Bartoszynski, Ireneusz Reclaw, Not Every $\gamma$-Set is Strongly Meager.Andreas Blass, Reductions Between Cardinal Characteristics of the Continuum.Claude Laflamme, Filter Games and Combinatorial Properties of (...) Strategies.R. Daniel Mauldin, Analytic non-Borel Sets Modulo Null Sets.Janusz Pawlikowski, Laver's Forcing and Outer Measure.Marion Scheepers, Meager Sets and Infinite Games.Saharon Shelah, On Some Problems in General Topology.Saharon Shelah, Remarks on $\aleph_1$-CWH not CWH First Countable Spaces.Juris Steprans, Cardinal Invariants Associated with Hausdorff Capacities. (shrink)

W artykule podjęto kwestię władzy jako istotnej kategorii występującej w komiksach serii „Tytus, Romek i A’Tomek” H. J. Chmielewskiego. Relacje władzy i przywództwa są tworzone i reprodukowane w mikrośrodowisku, jaką tworzy trzech głównych bohaterów zrzeszonych w specyficzny rodzaj koleżeńskiej grupy: tworzą oni zastęp harcerski funkcjonujący przy szkole podstawowej. Stabilność tej grupy podtrzymywana jest przez ramy instytucjonalne, w których funkcjonują bohaterowie. W kategoriach analitycznych władza jest tu rozumiana na sposób neoweberowski (podejście transformacyjne) oraz poststrukturalistyczny (władza dyscyplinarna). Szczególną rolę pełni tu (...) postać zastępowego A’Tomka: jest on liderem środowiskowym, autorytetem i przedstawicielem elit. (shrink)

We study a strengthening of the notion of a perfectly meager set. We say that a subset A of a perfect Polish space X is countably perfectly meager in X, if for every sequence of perfect subsets $\{P_n: n \in \mathbb N\}$ of X, there exists an $F_\sigma $ -set F in X such that $A \subseteq F$ and $F\cap P_n$ is meager in $P_n$ for each n. We give various characterizations and examples of countably perfectly meager sets. We prove (...) that not every universally meager set is countably perfectly meager correcting an earlier result of Bartoszyński. (shrink)

We give a new characterization of the cardinal invariant $\mathfrak {d}$ as the minimal cardinality of a family $\mathcal {D}$ of tall summable ideals such that an ultrafilter is rapid if and only if it has non-empty intersection with all the ideals in the family $\mathcal {D}$. On the other hand, we prove that in the Miller model, given any family $\mathcal {D}$ of analytic tall p-ideals such that $\vert \mathcal {D}\vert <\mathfrak {d}$, there is an ultrafilter $\mathcal {U}$ which (...) is an $\mathscr {I}$ -ultrafilter for all ideals $\mathscr {I}\in \mathcal {D}$ at the same time, yet $\mathcal {U}$ is not a rapid ultrafilter. As a corollary, we obtain that in the Miller model, given any analytic tall p-ideal $\mathscr {I}$, $\mathscr {I}$ -ultrafilters are dense in the Rudin–Blass ordering, generalizing a theorem of Bartoszyński and S. Shelah, who proved that in such model, Hausdorff ultrafilters are dense in the Rudin–Blass ordering. This theorem also shows some limitations about possible generalizations of a theorem of C. Laflamme and J. Zhu. (shrink)

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the meager ideal was (...) established by Bartoszyński and Kada. (shrink)

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is not (...) null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties. (shrink)

We prove the following theorems: (1) If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s 0 . (2) If X has Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. (3) If X has strong measure zero and Hurewicz's covering property then its algebraic sum with any set in APC ' is a (...) set in APC '. (APC ' is included in the class of sets always of first category, and includes the class of strong first category sets.) These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the γ-property and of a first category set is a first category set, and Bartoszynski and Judah's characterization of SR M -sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts. (shrink)

Tom specjalny „Avantu” poświęcamy twórczości komiksowej Henryka Jerzego Chmielewskiego, nazywanego Papciem Chmielem (także Dziadkiem Chmielem) oraz głównym postaciom serii komiksów „Tytus, Romek i A’Tomek” (dalej: „TRiA”). Wydawana od 1966 roku w formie osobnych albumów seria liczy 31 ksiąg, stając się najdłużej wydawanym w Polsce cyklem, który zajął poczesne miejsce w historii polskiego komiksu. Początkowo przy-gody Tytusa jawiły się jako wejście w krainę wyobraźni dziecięcej (księgi III-XVIII), z czasem jednak Chmielewski przeszedł do doraźnych komentarzy na temat zmieniającej się polskiej rzeczywistości (...) (od Księgi XX). Zmiana ta nastąpiła w momencie transformacji systemowej w Polsce. Przełom 1989/1990 to okres, w którym „Tytusy” zaczęły być wznawiane przez autora w unowo-cześnionej szacie graficznej (część książeczek narysowano na nowo) i zmienionej formie fabularnej: pojawiły się nowe rysunki, przearanżowano część starszych, dodano wypowiedzi bohaterów. W tym okresie Chmielewski nie wydawał książeczek z nowymi przygodami Tytusa; te zaczęły się pojawiać od 1992 roku, a autor zaczął opisywać w nich zmiany zachodzące w Polsce. Po przełomie politycznym 1989/1990 magiczne i osadzone w wyobraźni wątki zestawione zostały z tym, co w aktualnej polskiej rzeczywistości społecznej wydawało się Papciowi zajmujące, a często – niepokojące lub wymagające interwencji. Tytus zatem stał się członkiem afiliowanego przy NATO oddziału wojskowego, walczył z handlarzami narkotyków, rysował graffiti, mistrzowsko jeździł na desce, zwalczał bądź to chuliganów na meczach piłkarskich, bądź to gangi przestępcze, czy też intensywnie poznawał internet. (shrink)