Luzin Set Research Articles

First-countable Lindelöf scattered spaces

We study the class of first-countable Lindelöf scattered spaces, or “FLS” spaces. While every T3 FLS space is homeomorphic to a scattered subspace of Q, the class of T2 FLS spaces turns out to be surprisingly rich. Our investigation of these spaces reveals close ties to Q-sets, Lusin sets, and their relatives, and to the cardinals b and d. Many natural questions about FLS spaces turn out to be independent of ZFC.We prove that there exist uncountable FLS spaces with scattered height ω. On the other hand, an uncountable FLS space with finite scattered height exists if and only if b=ℵ1. We prove some independence results concerning the possible cardinalities of FLS spaces, and concerning what ordinals can be the scattered height of an FLS space. Several open problems are included.

Ramsey theory over partitions III: Strongly Luzin sets and partition relations

The strongest type of coloring of pairs of countable ordinals, gotten by Todorčević from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size ℵ 1 \aleph _1 . In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpiński a hundred years ago.

P-points, MAD families and Cardinal Invariants

P-points, MAD families and Cardinal Invariants

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.

NONMEASURABLE SETS AND UNIONS WITH RESPECT TO TREE IDEALS

AbstractIn this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$ , $m_0$ , $l_0$ , $cl_0$ , $h_0,$ and $ch_0$ . We show that there exists a subset of the Baire space $\omega ^\omega ,$ which is s-, l-, and m-nonmeasurable that forms a dominating m.e.d. family. We investigate a notion of ${\mathbb {T}}$ -Bernstein sets—sets which intersect but do not contain any body of any tree from a given family of trees ${\mathbb {T}}$ . We also obtain a result on ${\mathcal {I}}$ -Luzin sets, namely, we prove that if ${\mathfrak {c}}$ is a regular cardinal, then the algebraic sum (considered on the real line ${\mathbb {R}}$ ) of a generalized Luzin set and a generalized Sierpiński set belongs to $s_0, m_0$ , $l_0,$ and $cl_0$ .

A note on the Borel types of some small sets

Abstract The Borel types of some classical small subsets of the real line are considered. In particular, under Martin’s axiom it is shown that there are at least 𝐜 + {{\mathbf{c}}^{+}} pairwise incomparable Borel types of generalized Luzin sets (resp. of generalized Sierpiński sets), where 𝐜 {{\mathbf{c}}} stands for the cardinality of the continuum.

A decomposition theorem for locally compact groups

Abstract We prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalised notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrisable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type.

THE ONTO MAPPING OF SIERPINSKI AND NONMEAGER SETS

AbstractThe principle (*) of Sierpinski is the assertion that there is a family of functions $\left\{ {{\varphi _n}:{\omega _1} \to {\omega _1}|n \in \omega } \right\}$ such that for every $I \in {[{\omega _1}]^{{\omega _1}}}$ there is n ε ω such that ${\varphi _n}[I] = {\omega _1}$. We prove that this principle holds if there is a nonmeager set of size ω1 answering question of Arnold W. Miller. Combining our result with a theorem of Miller it then follows that (*) is equivalent to $non\left( {\cal M} \right) = {\omega _1}$. Miller also proved that the principle of Sierpinki is equivalent to the existence of a weak version of a Luzin set, we will construct a model where all of these sets are meager yet $non\left( {\cal M} \right) = {\omega _1}$.

Measurability properties of certain paradoxical subsets of the real line

Abstract The paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure μ on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.

Some properties of [formula omitted]-Luzin sets

In this paper we consider a notion of I-Luzin set which generalizes the classical notion of Luzin set and Sierpiński set on Euclidean spaces. We show that there is a translation invariant σ-ideal I with Borel base for which I-Luzin set can be I-measurable. If we additionally assume that I has the Smital property (or its weaker version) then I-Luzin sets are I-nonmeasurable. We give some constructions of I-Luzin sets involving additive structure of Rn. Moreover, we show that if c is regular, L is a generalized Luzin set and S is a generalized Sierpiński set then the complex sum L+S belongs to Marczewski ideal s0.

On some Problem of Sierpiński and Ruziewicz Concerning the Superposition of Measurable Functions. Microscopic Hamel Basis

Abstract S. Ruziewicz and W. Sierpiński proved that each function f : ℝ → ℝ can be represented as a superposition of two measurable functions. Here, a strengthening of this theorem is given. The properties of Lusin set and microscopic Hamel bases are considered

Selective covering properties of product spaces

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper):(1)Every product of a concentrated space with a Hurewicz S1(Γ,O) space satisfies S1(Γ,O). On the other hand, assuming the Continuum Hypothesis, for each Sierpiński set S there is a Luzin set L such that L×S can be mapped onto the real line by a Borel function.(2)Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger.(3)Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy.(4)Assuming d=ℵ1, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers. A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than add(N), the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy (u<g) with cov(M)>ℵ1.Our results improve upon and unify a number of results, established earlier by many authors.

Elementary chains and compact spaces with a small diagonal

It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of substructures. We prove that ccc subspaces of such spaces have countable π-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.

Weakly infinite dimensional subsets of [formula omitted

The Continuum Hypothesis implies an Erdös–Sierpiński like duality between the ideal of first category subsets of R N , and the ideal of countable dimensional subsets of R N . The algebraic sum of a Hurewicz subset – a dimension theoretic analogue of Sierpiński sets and Lusin sets – of R N with any compactly.

Menger subsets of the Sorgenfrey line

A space X is said to have the Menger property if for every sequence {u n : n ∈ ω} of open covers of X, there are finite subfamilies V n ⊂ U n (n, ∈ ω) such that U n∈ω V n is a cover of X. Let i: S → R be the identity map from the Sorgenfrey line onto the real line and let X S = i -1 (X) for X ⊂ ℝ. Lelek noted in 1964 that for every Lusin set L in ℝ, L S has the Menger property. In this paper we further investigate Menger subsets of the Sorgenfrey line. Among other things, we show: (1) If X S has the Menger property, then X has Marczewski's property (s 0 ). (2) Let X be a zero-dimensional separable metric space. If X has a countable subset Q satisfying that X A has the Menger property for every countable set A C X Q, then there is an embedding e: X → R such that e(X) S has the Menger property. (3) For a Lindelof subspace of a real GO-space (for instance the Sorgenfrey line), total paracompactness, total metacompactness and the Menger property are equivalent.

On a Theorem of Banach and Kuratowski and $K$-Lusin Sets

In a paper of 1929, Banach and Kuratowski proved-assuming the continuum hypothesis-a combinatorial theorem which implies that there is no non-vanishing σ-additive finite measure μ on R which is defined for every set of reals. It will be shown that the combinatorial theorem is equivalent to the existence of a K-Lusin set of size 2 N 0 and that the existence of such sets is independent of ZFC + ¬CH.

Terminal notions in set theory

In mathematical practice certain formulas φ(x) are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ(x), namely “ x is a Ramsey ultrafilter”, “ x is a free Souslin tree”, “ x is an extendible strong Lusin set” and “ x is a good diamond sequence”.

Strongly meager sets and their uniformly continuous images

We prove the following theorems: (1) Suppose that f; 2W 2W is a continuous function and X is a Sierpiiiski set. Then (A) for any strongly set Y, the image f[X Y] is an so-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (see [N], [NSW], [S]) in which properties (mainly, the algebraic sum) of certain singular subsets of the real line R and of the Cantor set 2' were investigated. Throughout the paper, by a set of real numbers we mean a subset of 2' and by + we denote the standard modulo 2 coordinatewise addition in 2W. Let us also assume that a measure zero (or negligible) set always denotes a Lebesgue set. We apply the following definition of sets of real numbers. Definition 1. An uncountable set X is said to be a Luzin (respectively, Sierpin'ski) set iff for each meager (respectively, zero) set Y, XnY is at most countable. We say that a set X is of strong (respectively, strongly meager) iff for each meager (respectively, zero) set Y, X Y 7& 2'. Remark 1. It is well known (see [M] for example) that every Luzin set is strongly zero. Quite recently J. Pawlikowski proved that each Sierpin'ski set must be strongly meager as well (see [P]). Let us recall that a set X is called an so-set (or Marczewski set) iff for each perfect set P one can find a perfect set QCP that is disjoint from X. M. Scheepers showed in [S] that for a Sierpin'ski set X and a strong set Y, X Y is an so-set. Later, in [NSW] it was proven that this also holds when X is strongly meager. We have the following functional version of the M. Scheepers' result. Theorem 1. Let X be a Sierpin'ski set and let Y be a strong set. Assume also that f: 2' -* 2W is a continuous function. Then the image f[X Y] is an so-set. Received by the editors July 16, 1998 and, in revised form, September 9, 1998 and March 10, 1999. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15.

Special subsets of cf( μ) μ, Boolean algebras and Maharam measure algebras

The original theme of the paper is the existence proof of “there is η ̄ =〈η α:α<λ〉 which is a ( λ, J)-sequence for I ̄ =〈I i:i<δ〉 , a sequence of ideals”. This can be thought of as a generalization to Luzin sets and Sierpinski sets, but for the product ∏ i<δ dom(I i) , the existence proofs are related to pcf. The second theme is when does a Boolean algebra B have a free caliber λ (i.e., if X⫅ B and | X|= λ, then for some Y⫅ X with | Y|= λ and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product of free Boolean algebras, and κ-cc Boolean algebras. A central case is λ=(ℶ ω) + , or more generally, λ= μ + for μ strong limit singular of “small” cofinality. A second one is μ= μ < κ < λ<2 μ ; the main case is λ regular but we also have things to say on the singular case. Lastly, we deal with ultraproducts of Boolean algebras in relation to irr(-) and s(-) etc.

Lusin sets

Lusin sets