Not every gamma-set is strongly meager
英文摘要:
We present two constructions of gamma-sets which are large in sense of category.
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Tomek Bartoszynski, Ireneusz Reclaw
We present two constructions of gamma-sets which are large in sense of category.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Every null additive set of reals is meager additive
我们证明了每个零可加集都是贫乏可加的,其中:(1)一个集合X subseteq 2 ^omega是零可加的,如果对于每个勒贝格零集A subseteq 2 ^omega,X+A也是零的;(2)我们说X subseteq 2 ^omega是贫乏可加的,如果对于每个贫乏的A subseteq 2 ^omega,X+A也是贫乏的。
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))
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 prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager 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.
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.
A $*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $*$-reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish $G$-spaces which is generically preserved under $*$-reductions. For every natural number $n$ we define a free action of $S_{\infty}$ whose dimension is $n$ on every invariant Baire measurable non-meager set. We also show that the $S_{\infty}$-space which induc…
Following Darji, we say that a Borel subset $B$ of an abelian Polish group $G$ is Haar meager if there is a compact metric space $K$ and a continuous function $f : K \to G$ such that the preimage of the translate, $f^{-1}(B+g)$ is meager in $K$ for every $g \in G$. The set $B$ is called strongly Haar meager if there is a compact set $C \subseteq G$ such that $(B+g) \cap C$ is meager in $C$ for every $g \in G$. The main open problem in this area is Darji's question asking whether these two notions are the same. Even though there have been several partial results suggesting a positive answer, in…
Nicely generated and chaotic ideals
我们证明了如果实直线是kappa稀疏集的不交并,使得每个稀疏集都包含在它们的可数并中,则kappa = omega_1。这回答了J. Cichon提出的问题。我们还证明了两个定理,大致说,任何试图产生的同构型的微薄的理想在科恩真正的和随机的实际扩展必须失败。我们所有的结果也都适用于用“null”代替“meager”。
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.
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' ".
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。
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}…
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…
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。
In [arXiv:1605.02261] Rosłanowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with $ZFC$ that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper we give affirmative answers for the general case.
Miklos Laczkovich asked if there exists a Haussdorff (or even normal) space in which every subset is Borel yet it is not meager. The motivation of the last condition is that under MA_kappa every subspace of the reals of cardinality kappa has the property that all subsets are F_sigma, however Martin's axiom also implies that these subsets are meager. Here we answer Laczkovich' question.
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.
Souslin's Hypothesis and Convergence in Category
从Baire空间X到实数的函数序列f_n:X -> R称为在范畴内收敛当且仅当每个子序列都有一个子序列在除贫乏集外的所有集上收敛。证明了如果存在一棵Souslin树,则存在一个非原子Baire空间X,使得每一个在范畴内收敛的序列处处收敛于一个Comager集.这回答了Wagner和Wilczynski的问题,可测函数序列的收敛,Acta Math Acad Sci Hung 36(1980),125-128。
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 .
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.
We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the randomly ordered rational Urysohn space.
The Category Dichotomy for Ideals
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We construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f:R-->R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of D. Banaszewski. We also show that every extendable function g:R-->R with a dense graph satisfies the following stronger version of the SCIVP property: for every a<b and every perfect set K between g(a) and g(b) there is a perfect subset C of (a,b) such that g[C] subset K and g|C is continuous strictly increasing. This property is…
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In this paper we show that it is relatively consistent with ZFC that every gamma-set is countable while not every strong measure zero set is countable. This answers a question of Paul Szeptycki. A set is a gamma-set iff every omega-cover contains a gamma-subcover. An open cover is an omega-cover iff every finite set is covered by some element of the cover. An open cover is a gamma-cover iff every element of the space is in all but finitely many elements of the cover. Gerlits and Nagy proved that every gamma-set has strong measure zero. We also show that is consistent that every strong gamma-se…
The automorphism group of the random poset does not admit a generic pair
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Khan and Miller proved that for every computable non decreasing unbounded function $h\in ω^ω$ (henceforth order function), if $h$ is sufficiently large, then there exists a $DNR_h$ that is of minimal degree. Where $h$ has to satisfy $\lim_{n\rightarrow\infty} h(n)/(2^{k\cdot \prod_{m<n}h(m)})=\infty$ for all $k>0$. Their core argument is that we can thin the tree by a factor of $2^j$ to make $j$ Turing functional split. We improve their result by reducing this factor to $j$. Thus we show that for every order function $h$ with $\lim_{n\rightarrow\infty} h(n)/( \prod_{m<n}h(m))^k=\infty$ for all…
A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$
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In this paper we prove that if there is a Borel-cover gamma-set of cardinality the continuum, then there is one which is not hereditary. A set of reals X is a Borel-cover gamma-set iff for every countable family of Borel sets which is an omega-cover contains a gamma-cover. This is also denoted S_1(Borel_omega, Borel_gamma). This result partially answers a question of Bartoszynski and Tsaban. Tsaban points out that it also gives an example of set which is both a gamma-set and sigma-set but is not hereditarily gamma, which answers a question of Bukovsky, Reclaw, and Repicky.
Rigid many-one degrees contain infinite antichains of $1$-degrees
本文证明了m-刚性的集合其许多一度包含无限个1度的反链,并展示了该结论在随机实数中的应用。
A subset X of the Cantor space, 2^ω, is a lambda-prime-set iff for every countable subset Y of the Cantor space Y is relatively G-delta in X union Y. In this paper we prove two forcing results about lambda-prime-sets. First we show that it is consistent that every lambda-prime-set is a gamma-set. Secondly we show that is independent whether or not every dagger-lambda-set is a lambda-prime-set.
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For $X$ a separable metric space define $\pp(X)$ to be the smallest cardinality of a subset $Z$ of $X$ which is not a relative $\ga$-set in $X$, i.e., there exists an $\om$-cover of $X$ with no $\ga$-subcover of $Z$. We give a characterization of $\pp(2^\om)$ and $\pp(\om^\om)$ in terms of definable free filters on $\om$ which is related to the psuedointersection number $\pp$. We show that for every uncountable standard analytic space $X$ that either $\pp(X)=\pp(2^\om)$ or $\pp(X)=\pp(\om^\om)$. We show that both of following statements are each relatively consistent with ZFC: (a) $\pp=\pp(\om…
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