Tomek Bartoszynski
A set of reals A is called perfectly meager if A \cap P is meager in P, for every perfect set P. Marczewski asked if the product of perfectly meager sets is perfectly meager. In the paper it is shown that it is consistent that the answer to this question is positive. (It is known that it is also consistent that the answer is negative (Reclaw))
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We will show that there is no ZFC example of a set distinguishing between universally null and perfectly meager sets.
We study a strengthening of the notion of a perfectly meager set. We say that 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 $F_σ$-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.
Countably perfectly meager and countably perfectly null sets
本文通过引入可数完全贫乏集和可数完全零集以强化普遍贫乏集与普遍零集概念，并在连续统假设弱化条件下证明了二者不等价。
We show that the Dual Borel Conjecture implies that ${\mathfrak d}> \aleph_1$ and find some topological characterizations of perfectly meager and universally meager sets.
We show that the following are consistent with ZFC: 1. Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).
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.
We study the ideal of meager sets and related ideals.
A set X subseteq R is strongly meager if for every measure zero set H, X+H not= R. Let SM denote the collection of strongly meager sets. We show that assuming CH, SM is not an ideal.
We show that the set of the ground-model reals has strong measure zero (is strongly meager) after adding a single Cohen real (random real). As consequence we prove that the set of the ground-model reals has strong measure zero after adding a single Hechler real.
We develop a theory of \emph{sharp measure zero} sets that parallels Borel's \emph{strong measure zero}, and prove a theorem analogous to Galvin-Myscielski-Solovay Theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^ω$ is meager-additive if and only if it is $\mathcal E$-additive; if $f:2^ω\to2^ω$ is continuous and $X$ is meager-additive, then so is $f(X)$.
The paper contains two results pointing to the lack of symmetry between measure and category. Assume CH. There exists a strongly meager subset of the Cantor set that can be mapped onto the Cantor set by a uniformly continuous function. (It is well known that uniformly continuous image of a strongly null set is strongly null). A ZFC version of this result is also given.
Combinatorics of translations of meager and closed measure zero sets
研究贫集与闭测度零集平移的组合性质，探讨相关Borel猜想，获得贫理想可加性新刻画，并证明若干基数不变量关系。
Player ONE chooses a meager set and player TWO, a nowhere dense set per inning. They play $ω$ many innings. ONE's consecutive choices must form a (weakly) increasing sequence. TWO wins if the union of the chosen nowhere dense sets covers the union of the chosen meager sets. A strategy for TWO which depends on knowing only the uncovered part of the most recently chosen meager set is said to be a remainder strategy. Theorem (among others): TWO has a winning remainder strategy for this game played on the real line with its usual topology.
We present two constructions of gamma-sets which are large in sense of category.
We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszyński and Judah.
A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff dimension. We investigate these sets and show that in 2^ω they coincide, respectively, with strongly null, meager-additive, T' and null-additive sets. Some consequences: A subset of 2^ω is meager-additive if and only if it is E-additive; if f:2^ω->2^ω is continuous and X is meager-additive, then so is f(X), and likewise for null-additive and T'-sets.
We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.
Cardinal invariants associated with the combinatorics of the uniformity number of the ideal of meager-additive sets
本文综述了与贫瘠可加集理想均匀数相关的基数不变量 $\mathfrak{b}_b^\mathsf{eq}$ 及其对偶 $\mathfrak{d}_b^\mathsf{eq}$，探讨它们与 Cichoń 图中基数的联系，并提出若干公开问题。
We study the relationship between the sigma-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.
Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\leqω$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset $F$ of $X$) such that $F$ is $R$-free for every $R\in\mathcal{R}$. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on $ω$, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting "$|\mathcal{R}|\le…
We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which …
We show that if the real line is the disjoint union of kappa meager sets such that every meager set is contained in a countable union of them, then kappa = omega_1. This answers a question addressed by J. Cichon. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. All our results hold for "meager" replaced by "null", as well.
Universally meager sets in the Miller model and similar ones
研究在Miller模型及其他模型中普遍渺小集的大小限制，证明强测度零集的大小至多为ω₁，并探讨相关覆盖性质与空间结构的关系。
A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $μ$ and a Borel set $B$ containing $X$ such that $μ(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f : K \to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h \in G$. We calculate (in $ZFC$) the four cardinal invariants ($\rm add$, $\rm cov$, $\rm non$, $\rm cof$) of these two $σ$-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb{Z}…
We show that some set-theoretic assumptions (for example Martin's Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets in (in R^n). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the sigma-algebra of sets with the property of Baire.
Directed schemes of ideals and cardinal characteristics, I: the meager additive ideal
本文引入方向理想方案以刻画实数上的特殊理想，提出方向方案$\vec{\mathcal{M}}$用于描述meager加法理想，探讨其组合学及基数特征，并证明cov(NA)<c与cof(MA)<non(SN)的一致性。
We consider ``meager analogues'' of classical covering properties of Menger, Hurewicz and Rothberger. We show that Borel images of sets having ``our'' covering properties have these classical covering properties.
We show that (1) If ZF is consistent then the following theory is consistent "ZF + DC(omega_{1}) + Every set of reals has Baire property" and (2) If ZF is consistent then the following theory is consistent "ZFC + `every projective set of reals has Baire property' + `any union of omega_{1} meager sets is meager' ".
We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.
A bijection from R to R is called an Erdos-Sierpinski mapping if it maps meager sets onto the first category sets and vice versa. We show that there is no Erdos-Sierpinski mapping preserving addition.
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. 2. Using Scheepers' notation for selection principles: Sfin(Omega,Omega^gp)\cap S1(O,O)=S1(Omega,Omega^gp), and Borel's Conjecture for S1(Omega,Omega) (or just S1(Omega,Omega^gp)) implies Borel's Conjecture. These results extend results of Scheepers and Mill…
It is proved that no non-meager subspace of the space $[ω]^ω$ equipped with the Ellentuck topology does admit a Kuratowski partition, that is such a subset cannot be covered by a family $\mathfrak{F}$ of disjoint relatively meager sets such that $\bigcup\mathfrak{F}'$ has the Baire property (relatively) for every subfamily $\mathfrak{F}'\subseteq\mathfrak{F}$. It is also shown that the existence of Kuratowski partitions is not a metric problem. Some remarks concerning continuous restrictions of functions with domain in the Ellentuck space are made.
The main goal of this paper is to generalize several results concerning cardinal invariants to the statements about the associated families of sets. We also discuss the relationship between the additive properties of sets and their Borel images. Finally, we present estimates for the size of the smallest set which is not strongly meager.
We use the method of norms on possibilities to answer a question of Kunen and construct a ccc sigma --ideal on 2^omega with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection.
We introduce two new classes of special subsets of the real line: the class of perfectly null sets and the class of sets which are perfectly null in the transitive sense. These classes may play the role of duals to the corresponding classes on the category side. We investigate their properties and, in particular, we prove that every strongly null set is perfectly null in the transitive sense, and that it is consistent with ZFC that there exists a universally null set which is not perfectly null in the transitive sense. Finally, we state some open questions concerning the above classes. Althoug…
We ask whether $\mathbf{Δ^1_2}$ or $\mathbf{Σ^1_2}$ equivalence relations with $I$-small classes for $I$ a $σ$-ideal must have perfectly many classes. We show that for a wide class of ccc $σ$-ideals, a positive answer for $\mathbf{Δ^1_2}$ equivalence relations is equivalent to the $I$-measurability of $\mathbf{Δ^1_2}$ sets. However, the analogous statement for $\mathbf{Σ^1_2}$ equivalence relations is false: $\mathbf{Σ^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if $\mathbf{Δ^1_2}$ sets have the Baire property.
On Star operation and some ideals on the Baire space
本文研究了Baire空间上的几个σ-理想，引入并分析了理想G和SMZ⁺，建立了它们与经典理想的结构关系，计算了M₋的基数不变量，并证明了星运算下的对偶关系，最后证明了M₋的树二分定理。
For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor space for which one may force aleph_epsilon translations with intersections of size 2i, but such that it has no perfect set of such translations in any ccc extension. These sets have uncountably many translations with intersections of size 2i in ZFC, so this answers Problem 3.4 of arxiv:1711.04058 .
For a Polish group G let cov_G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(meager) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups cov_G=cov(meager). For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces. Most re…
We will prove that there exists a model of ZFC+``c= omega_2'' in which every M subseteq R of cardinality less than continuum c is meager, and such that for every X subseteq R of cardinality c there exists a continuous function f:R-> R with f[X]=[0,1]. In particular in this model there is no magic set, i.e., a set M subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R-> R .
We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no perfect set P such that |(eta+B) cap (nu+B)|>5 for any distinct eta,nu from P. This answers two questions from our previous works.
We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups $(H,+)$. In particular, we show that if $α<ω_1$ and $4\leq k<ω$, then there is a ccc forcing notion adding a $Σ^0_2$ set $B\subseteq H$ which has $\aleph_α$ many pairwise $k$--overlapping translations but not a perfect set of such translations.
For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134
Let ${\bf x}=(x_n)_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\|\sum_{n\in F}x_n\|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B({\bf x})$ iff there is a non pathological lower semicontinuous submeasure $φ$ on $\mathbb{N}$ such that $\mathcal{I} =FIN(φ)=\{A\subseteq \mathbb{N}: \;φ(A)<\infty\}$. We address the questions of when $FIN(φ)$ is a tall ideal and has a Borel selector. W…
We recall some classical results relating normality and some natural weakenings of normality in $Ψ$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $λ$-sets and $σ$-sets. We introduce a new class of special sets of reals which corresponds the corresponding almost disjoint family of branches being $\aleph_0$-separated. This new class fits between $λ$-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $\mathcal A$ being potentially almost-normal (pseudonormal), in the sense that $\mathcal A$ is almo…
Uncountable sets and an infinite linear order game
本文针对 Baker 博弈中 Player II 能否在一般线性序的不可数集上拥有必胜策略的问题给出肯定回答，证明了对任意无限基数均可构造使 Player II 对所有收益集必胜的稠密线性序，并指出了推广该刻画的困难。
Topological Erdős similarity conjecture and strong measure zero sets
本文证明实数集上拓扑普适集等价于强零测集，并推广至局部紧 Polish 群，从而确立不可数拓扑普适集存在性对 ZFC 的独立性，还提出全测度普适 Erdős 相似性猜想。
We investigate families of subsets of $ω$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $ω$. More precisely, we study the following topics and questions: 1) Examples of projective ideals. 2) We prove the following generalization of a result due to J. Brendle: If $V\subseteq W$ are transitive models, $ω_1^W\subseteq V$, $\mathcal{P}(ω)\cap V\not = \mathcal{P}(ω)\cap W$, and $\mathcal{I}$ is an analytic or coanalytic ideal coded in $V$, then there is an $\mathcal{I}$-almost disjoint refinement ($\mathcal{I}$-ADR) of $\mathcal{I}^+\cap V$ in $W$…
This paper deals with different concepts for characterizing the size of mathematical objects. A game theoretic investigation and generalization of two size concepts, which can both be formulated in topological terms, is provided: the so called "Baire category" and the "$σ$-category". This is mainly done by means of (generalized) Banach-Mazur games using the Axiom of determinacy (the inconsistency of AC and AD is reflected in the beginning and a weaker form of AC is chosen for proofs). Analogue versions of Cantor-Bendixson and Heine-Borel are proofed as well as some definability results. Such s…
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