Meager Sets Research Articles

A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF

AbstractFor any subset $Z \subseteq {\mathbb {Q}}$ , consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$ . Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$ , we show that if Z is not thin in ${\mathbb {Q}}$ , then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ . Here, thin and meager both mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $ -definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

Lipschitz-Free Spaces over Cantor Sets and Approximation Properties

Let K=2^mathbb {N} be the Cantor set, let mathcal {M} be the set of all metrics d on K that give its usual (product) topology, and equip mathcal {M} with the topology of uniform convergence, where the metrics are regarded as functions on K^2. We prove that the set of metrics din mathcal {M} for which the Lipschitz-free space mathcal {F}(K,d) has the metric approximation property is a residual F_{sigma delta } set in mathcal {M}, and that the set of metrics din mathcal {M} for which mathcal {F}(K,d) fails the approximation property is a dense meager set in mathcal {M}. This answers a question posed by G. Godefroy.

On countably perfectly meager and countably perfectly null sets

We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set.We say that a subset A of a perfect Polish space X is countably perfectly meager (respectively, countably perfectly null) in X, if for every perfect Polish topology τ on X, giving the original Borel structure of X, A is covered by an Fσ-set F in X with the original Polish topology such that F is meager with respect to τ (respectively, for every finite, non-atomic, Borel measure μ on X, A is covered by an Fσ-set F in X with μ(F)=0).We prove that if 2ℵ0≤ℵ2, then there exists a universally meager set in 2N which is not countably perfectly meager in 2N (respectively, a universally null set in 2N which is not countably perfectly null in 2N).

Set of Uniqueness for Cantorvals

The main result is that the celebrated Guthrie–Nymann’s Cantorval has comeager set of uniqueness. On the other hand many other Cantorvals have meager set of uniqueness.

Almost continuous Sierpiński–Zygmund functions under different set-theoretical assumptions

A function f:mathbb {R}rightarrow mathbb {R} is: almost continuous in the sense of Stallings, fin textrm{AC}, if each open set Gsubset mathbb {R}^2 containing the graph of f contains also the graph of a continuous function g:mathbb {R}rightarrow mathbb {R}; Sierpiński–Zygmund, fin textrm{SZ} (or, more generally, fin textrm{SZ}(textrm{Bor})), provided its restriction frestriction M is discontinuous (not Borel, respectively) for any Msubset mathbb {R} of cardinality continuum. It is known that an example of a Sierpiński–Zygmund almost continuous function f:mathbb {R}rightarrow mathbb {R} (i.e., an fin textrm{SZ}cap textrm{AC}) cannot be constructed in ZFC; however, an fin textrm{SZ}cap textrm{AC} exists under the additional set-theoretical assumption {{,textrm{cov},}}(mathcal {M})=mathfrak {c}, that is, that mathbb {R} cannot be covered by less than mathfrak {c}-many meager sets. The primary purpose of this paper is to show that the existence of an fin textrm{SZ}cap textrm{AC} is also consistent with ZFC plus the negation of {{,textrm{cov},}}(mathcal {M})=mathfrak {c}. More precisely, we show that it is consistent with ZFC+{{,textrm{cov},}}(mathcal {M})<mathfrak {c} (follows from the assumption that {{,textrm{non},}}(mathcal {N})<{{,textrm{cov},}}(mathcal {N})=mathfrak {c}) that there is an fin textrm{SZ}(textrm{Bor})cap textrm{AC} and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either {{,textrm{cov},}}(mathcal {M})=mathfrak {c} or {{,textrm{non},}}(mathcal {N})<{{,textrm{cov},}}(mathcal {N})=mathfrak {c}, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński–Zygmund functions. Several open problems are also stated.

Haar null closed and convex sets in separable Banach spaces.

Haar null sets were introduced by Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, Darji defined a categorical version of Haar null sets, namely Haar meagre sets. The present paper aims to show that, whenever is a closed, convex subset of a separable Banach space, is Haar null if and only if is Haar meagre. We then use this fact to improve a theorem of Matoušková and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation of separable Banach lattices whose positive cone is not Haarnull.

Cichoń’s maximum without large cardinals

Cichoń’s diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $\sigma$-compact subsets of the irrationals. It is consistent that all entries of Cichoń’s diagram are pairwise different (apart from $\operatorname{add}\mathcal{M}$ and $\operatorname{cof}\mathcal{M}$, which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions. In this work, we show the consistency without such assumptions.

MEAGER-ADDITIVE SETS IN TOPOLOGICAL GROUPS

AbstractBy the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M.A set $X\subseteq \mathbb {R}$ is meager-additive if $M+X$ is meager for each meager set M. Recently a theorem on meager-additive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set $X\subseteq \mathbb {R}$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero.We investigate the validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants.

COUNTABLY PERFECTLY MEAGER SETS

AbstractWe 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.

On a strong generalized topology with respect to the outer Lebesgue measure

The classical density topology is the topology generated by the lower density operator connected with a density point of a set. One of the possible generalizations of the concept of a density point is replacing the Lebesgue measure by the outer Lebesgue measure. It turns out that the analogous family associated with such generalized density points is not a topology. In this case, one can prove that this family is a strong generalized topology. In the paper some properties of the strong generalized topology connected with density points with respect to the outer Lebesgue measure will be presented. Moreover, among others, some characterizations of the families of meager sets and compact sets in this space will be given.

The e-property of asymptotically stable Markov–Feller operators

In this work, we prove that any asymptotically stable Markov-Feller operator possesses the e-property everywhere outside at most a meagre set. We also provide an example showing that this result is tight. Moreover, an equivalent criterion for the e-property is proposed.

Higher-dimensional obstructions for star reductions

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 induces the equivalence relation $=^{+}$ of countable sets of reals is $\infty$-dimensional on every invariant Baire measurable non-meager set. We conclude that the orbit equivalence relations associated to all these actions are pairwise incomparable with respect to $*$-reductions.

The higher Cichoń diagram

For a strongly inacessible cardinal&nbsp;$\kappa $, we investigate the relationships between the following ideals: (1) the ideal of meager sets in the ${ \lt }\kappa $-box product topology, (2) the ideal of “null” sets in the sense o

On abstract generalized topological spaces generated by the density type operators

In the paper we concentrate on a generalized topological space generated by a density type operator on a measurable space. The properties of such generalized topological space are investigated. Moreover, the properties of nowhere dense sets, meager sets and compact sets in this generalized topological space are studied.

Special subsets of the generalized Cantor space and generalized Baire space

AbstractIn this paper, we are interested in parallels to the classical notions of special subsets in defined in the generalized Cantor and Baire spaces (2κ and ). We consider generalizations of the well‐known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ‐sets, γ‐sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less‐know classes like X‐small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver‐null sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems.

Meager sets, games and singular cardinals

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.

Haar null and Haar meager sets: a survey and new results

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 were already known for Haar null sets.

Lineability of the functions that are Sierpiński–Zygmund, Darboux, but not connectivity

Assuming that continuum $$\mathfrak {c}$$ is a regular cardinal, we show that the class $${{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn}\,}}$$ of all functions from $$\mathbb {R}$$ to $$\mathbb {R}$$ that are perfectly everywhere surjective (so Darboux) but not connectivity is $$\mathfrak {c}^+$$ -lineable, that is, that there exists a linear space of $$\mathbb {R}^\mathbb {R}$$ of cardinality $$\mathfrak {c}^+$$ that is contained in $$({{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn}\,}})\cup \{0\}$$ . Moreover, assuming additionally that $$\mathbb {R}$$ is not a union of less than $$\mathfrak {c}$$ -many meager sets, we prove $$\mathfrak {c}^+$$ -lineability of the class $${{\,\mathrm{SZ}\,}}\cap {{\,\mathrm{ES}\,}}{\setminus }{{\,\mathrm{Conn}\,}}$$ of Sierpinski-Zygmund everywhere surjective but not connectivity functions.

On G-transitive version of perfectly meager sets

We study the G-invariant version of perfectly meager sets (a generalization of the notion of AFC sets). We find the necessary and sufficient conditions for the inclusion AFC'G ⊆ I. In particular, we partially characterize for which groups G of automorphisms of the Cantor space every AFC'G set is Lebesgue null.

A Haar meager set that is not strongly Haar meager

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 → G such that the preimage of the translate f−1(B + g) is meager in K for every g ∈ G. The set B is called strongly Haar meager if there is a compact set C ⊆ G such that (B + g) ⋂ C is meager in C for every g ∈ 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 this paper we construct a counterexample. More specifically, we construct a Gδ set in ℤω that is Haar meager but not strongly Haar meager. We also show that no Fσ counterexample exists, hence our result is optimal.