Meager Sets Research Articles

A nonlinear Banach–Steinhaus theorem and some meager sets in Banach spaces

We establish a Banach–Steinhaus type theorem for nonlinear functionals of several variables. As an application, we obtain extensions of the recent results of Balcerzak and Wachowicz on some meager subsets of $L^1(\mu )\times L^1(\mu )$ and $c_0\time

Products of Special Sets of Real Numbers

We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: 1. The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measure zero/is strongly meager, respectively.

An Example of an Additive Almost Continuous Sierpiński-Zygmund Function

Assuming that the union of fewer than continuumly many meager sets does not cover the real line, we construct an example of an additive almost continuous Sierpi{\'n}ski-Zygmund function which has a perfect road at each point but which does not have the Cantor intermediate value property.

Saturation, Suslin trees and meager sets

We show, using a variation of Woodin’s partial order ℙ max , that it is possible to destroy the saturation of the nonstationary ideal on ω1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ1, saturation of the nonstationary ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal.

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2ω and a continuous functionF: 2ω → 2ω such thatF″ (X)=2ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false.

On coxeter functors for some classes of algebras generated by idempotents

We prove that a Borel separately Fσ-measurable function f: X×Y→R on the product of Polish spaces is a function of the first Baire class on the complement X×Y\M of a certain projectively meager set M⊂X×Y.

Properness Without Elementaricity

We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not neces- sarily elementary sub-models of some (H(χ), ∈). This leads to forcing notions which are reasonably definable. We present two specific properties mate- rializing this intuition: nep (non-elementary properness) and snep (Souslin non-elementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A ma- jor theme here is by iteration, but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.

Fubini properties for filter-related σ-ideals

Let M F be the σ-ideal of meager sets in the Ellentuck topology related to a filter F on the set of natural numbers. We prove that if G is a filter with the Baire property and F is an analytic, non-countably generated p-filter, then the pair 〈 M G , M F 〉 does not have Fubini Property. In fact, there is a Borel set on the plane such that its vertical sections are in M F and the complements of its horizontal sections are in M G . This partially answers a question of Recław [Topology Appl., submitted for publication].

Strongly meager sets of size continuum

We construct several models where there are no strongly meager sets of size continuum. In particular, there are no such sets in the Laver's model.

Some remarks on totally imperfect sets

We prove the following two theorems. Theorem 1. Let X be a strongly meager subset of 2 ω×ω . Then it is dual Ramsey null. We will say that a σ-ideal I of subsets of 2 satisfies the condition () iff for every X C 2, if ∀ f ∈ ω ↑ ω {g ∈ ω ω : ¬(f? g)} ∩ X ∈ I, then X ∈ I. Theorem 2. The σ-ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition ().

A scientific garden in a public space: the New York botanical garden, 1900–10

In November 1888, The New York Herald published a list of noteworthy botanic gardens, ranking Russia's St Petersburg garden as the ‘best in Europe’, ahead of such notable Old World landscapes as the Jardin des Plantes in Paris and the Royal Botanic Garden at Kew. In sharp contrast to these European showpieces, The Herald lamented, stood a meager set of us institutions, and of these, only Shaw's Garden in St Louis deserved comparison with horticultural and scientific masterpieces. Clearly, the Editors maintained, a great new American garden was needed, and New York City, a metropolis whose population approached one million, was just the setting for a new public space adorned with imposing structures of beauty. The new ‘floral study ground’ would in turn foster the ‘softening, civilizing education of both rich and poor’ within the city, The Herald predicted.1

On the Ramseyan properties of some special subsets of 2ωand their algebraic sums

AbstractWe prove the following theorems:1. IfX⊆ 2ωis aγ-set andY⊆2ωis a strongly meager set, thenX+Yis Ramsey null.2. IfX⊆2ωis aγ-set andYbelongs to the class ofsets, then the algebraic sumX+Yis anset as well.3. Under CH there exists a setX∈MGR* which is not Ramsey null.

Perfectly meager sets and universally null sets

We will show that there is no ZFC example of a set distinguishing between universally null and perfectly meager sets.

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that ${\mathfrak d}> \boldsymbol \aleph _1$ and find some topological characterizations of perfectly meager and universally meager sets.

Strongly meager and strong measure zero sets

In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID=""ID="" <E5>Mathematics Subject Classification (2000):</E5>&ensp;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

On perfectly meager sets

We show that it is consistent that the product of perfectly meager sets is perfectly meager.

Strongly meager sets of real numbers and tree forcing notions

We show that every strongly meager set has the l 0 l_0 - and the m 0 m_0 - property.

Properties of ideals on the generalized Cantor spaces

AbstractWe define a class of productiveσ-ideals of subsets of the Cantor space 2ω and observe that both σ-ideals of meagre sets and of null sets are in this class. From every productive σ-ideal we produce a σ-ideal of subsets of the generalized Cantor space 2κ. In particular, starting from meagre sets and null sets in 2ω we obtain meagre sets and null sets in 2ω, respectively. Then we investigate additivity, covering number, uniformity and cofinality of . For example, we show thatOur results generalizes those from [5].

On perfectly meager sets in the transitive sense

We prove that assuming c < N 2 one can always find a perfectly meager set, which is not perfectly meager in the transitive sense.

STRONGLY MEAGER SETS DO NOT FORM AN IDEAL

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.