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On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences

Joanna Jureczko
Published Online: 2018-07-04 | DOI: https://doi.org/10.1515/math-2018-0066

Abstract

In 2003 Bartoszyński and Halbeisen published the results on various equivalences of Kuratowski and Banach theorem from 1929 concerning some aspect of measure theory. They showed that the existence of the so called BK-matrix related to Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of cardinality continuum. On the other hand, in 1965 Efimov introduced the strong sequences method and using this method proved some well-known theorems in dyadic spaces. The goal of this paper is to show that the existence of such a K-Lusin set is equivalent to the existence of strong sequences of the same cardinality. Some applications of this results are also shown.

MSC 2010: 03E05; 03E10; 03E20; 03E35

1 Introduction

In the paper [1] Bartoszyński and Halbeisen proved some equivalences of the existence of K-Lusin sets of size 20 and showed that the existence of such a set is independent of ZFC+¬CH. Some of their results were inspired by a paper of Banach and Kuratowski [2] published in 1929, in which the authors solved a problem in measure theory concerning the existence of a non-vanishing σ-additive finite measure on the real line which is defined for every set of reals by using the following combinatorial result, (see [1]).

Banach and Kuratowski Theorem

Under the assumption of CH, there is an infinite matrix $\begin{array}{}{A}_{k}^{i}\end{array}$ ⊆ [0, 1] (where i, kω) such that

1. for each iω, [0, 1] = ⋃kω $\begin{array}{}{A}_{k}^{i}\end{array}$,

2. for each iω, if kkthen $\begin{array}{}{A}_{k}^{i}\end{array}$$\begin{array}{}{A}_{{k}^{\mathrm{\prime }}}^{i}\end{array}$ = ∅,

3. for every sequence k0, k1, …, ki, … of ω the set

$⋂i∈ω(A0i∪A1i∪...∪Akii)$

is at most countable.

The matrix from the theorem above is called in the literature as a BK-matrix.

Following [1] recall that if 𝓕 ⊆ ωω then

$λ(F)=min{η:∀g∈ωω|{f∈F:f⪯g}|<η},$

where f, gωω, fg iff |{nω : f(n) > g(n)}| < ω and

$l=min{λ(F):F⊆ωω∧|F|=c}.$

The crucial point in the Banach and Kuratowski proof, ([2], Theorem II), is the following result

Fact 1.1

The existence of BK-Matrix is equivalent to 𝔩 = ℵ1.

Obviously CH implies that 𝔩 = ℵ1. Among other theorems, Bartoszyński and Halbeisen showed that the existence of a K-Lusin set of cardinality 𝔠 is equivalent to 𝔩 = ℵ1, ([1], Lemma 2.3), and to the existence of a concentrated set of cardinality 𝔠, ([1], Proposition 3.4).

On the other hand, Efimov in [3] introduced the combinatorial method so called strong sequences method and used it for proving some well-known theorems in dyadic spaces (among others the Marczewski theorem on cellularity, the Shanin theorem on a calibre).

The goal of this paper is to show that the existence of a special kind of a strong sequence is equivalent to the existence of a K-Lusin set of cardinality 𝔠, which is an easy consequence of 𝔩 = ℵ1 and of the existence of a concentrated set of cardinality 𝔠. The results in this paper will be proved in a generalized version which generates further equivalences, especially between the existence of generalized strong sequences and the so called generalized BK-matrix as a kind of generalized independent family, (see [4,5,6]).

2 On the existence of generalized K-Lusin sets and generalized strong sequences

Let κω be a regular cardinal. In the set κκ we consider the following relation, if f, gκκ then gf iff |{ακ : g(α) > f(α)}| < κ.

A set Kκκ is said to be closed iff for all fK and for all gκκ if fg then gK. A closed set Kκκ is κ-compact iff there is a function fκκ such that K ⊆ {gκκ : gf}.

Definition 2.1

A set X of size 2κ is a generalized K-Lusin set if |XK| < 2κ for every κ-compact set Kκκ.

Now let 𝓕 ⊆ κ κ then

$λκ(F)=min{τ:∀g∈κκ|{f∈F:f⪯g}|<τ},lκ=min{λκ(F):F⊆κκ∧|F|=2κ}.$

Assuming GCH we obtain 𝔩κ = 2κ. It follows from the next lemma.

Lemma 2.2(GCH)

There exists a family 𝓕 ⊆ κκ of cardinality κ+ such that

$λκ(F)=2κ.$

Proof

Let

$κκ={gα:α<κ+}.$

We will construct a sequence {fακκ : α < κ+} such that

1. fβfα for β < α < κ+

2. gαfα for α < κ+

by transfinite recursion.

Assume that for α < κ+ the sequence {fβκκ : β < α} fulfilling (i) and (ii) has been defined. Since κ+ is regular, we have κκ ∖ {fβκκ : β < α} ≠ ∅.

Let α be a successor. Take gακκ and suppose that for each fακκ ∖ {fβκκ : β < α} at least one of the conditions (i) or (ii) does not hold.

If (ii) does not hold then |{fκκ : fgβ}| = 2κ for β < α . A contradiction. The argumentation for proving that (i) holds is similar.

If α is limit, then we take fα = ⋃β<α fβ. □

The following lemma is an easy consequence of the above definitions and Lemma 2.2.

Lemma 2.3

(GCH). The following are equivalent

1. 𝔩κ = 2κ

2. there exists a generalized K-Lusin set of cardinality 2κ.

Now we define a generalized strong sequence, (see also [6]). Let α be a cardinal number. We say that Aκκ is an α-directed set iff for each BA of cardinality less than α there exists fκκ such that gf for all gB.

Definition 2.4

Let α and η be cardinals. A sequence (Hϕ)ϕ<η, where HϕX, is called an α-strong sequence if:

• 1o

Hϕ is α-directed for all ϕ < η,

• 2o

HψHϕ is not α-directed for all ϕ < ψ < η.

The following result is true.

Theorem 2.5

(GCH). The following are equivalent

1. 𝔩κ = 2κ

2. there exists a κ+-strong sequence (Hξ)ξ<κ+ with |Hξ| < κ+ for all ξ < κ+.

Proof

By Lemma 2.2 there exists a family 𝓕 ⊆ κκ of cardinality κ+ with λκ(𝓕) = 2κ. Choose f0κκ arbitrarily. Let H0 = {f ∈ 𝓕 : ff0}. Hence |H0| < 2κ. Let H0 be the first element of a κ+-strong sequence.

Assume that the κ+-strong sequence (Hξ)ξ<ψ for ψ < κ+ such that

$Hξ={f∈F∖⋃η<ξHη:f⪯fξ},$

where fξκκ ∖ {fη : η < ξ}, fξfη, (fξfη means fηfξ and fξfη for η < ξ) has been defined.

Since ψ < κ+, |𝓕| = κ+ and κ+ is regular, we have that there exists fκκ ∖ {fξ : ξ < κ+} such that ffξ for any ξ < ψ. Name such f by fψ. Let Hψ = {f ∈ 𝓕 ∖ ⋃ξ < ψ Hξ : ffψ. Obviously |Hψ| < κ+, (because of our assumptions). By the construction HξHψ, ξ < ψ is not κ+-directed, (because fψfξ for any ξ < ψ).

Now let (Hξ)ξ<κ+ be a κ+-strong sequence with Hξκκ and |Hξ| < κ+ for each ξ < κ+. According to Definition 2.4 we have that for all ξ < ψ there exist hξHξ, hψHψ such that hξhψ. For any ψ < κ+ let

$Hψ′={hψ∈Hψ:ξ<ψ⇒∃hξ∈Hξhξ⊥hψ}.$

Since Hψ, ψ < κ+ is κ+-directed, there exists ψκκ such that hψ, for all hψ$\begin{array}{}{H}_{\psi }^{\prime }\end{array}$ . Thus we can describe Hψ as follows

$Hψ={f∈κκ:f⪯h¯ψ}.$

Let 𝓕 = ⋃{Hξ : ξ < κ+}. For any gκκ consider {f ∈ 𝓕 : fg}. Suppose that there exists goκκ such that |{f ∈ 𝓕 : fg0}| = κ+. But by definition of 𝓕, there exists ξ0 such that {f ∈ 𝓕 : fg0} ⊆ Hξ0. Hence |Hξ0| = κ+. A contradiction. □

The next corollary follows from Theorem 2.5 and Lemma 2.3.

Corollary 2.6

(GCH). The following are equivalent

1. there exists a generalized K-Lusin set of cardinality κ+

2. there exists an κ+-strong sequence (Hξ)ξ<κ+, Hξκκ with |Hξ| < κ+ for any ξ < κ+.

For a cardinal α we introduce the following notation, (see [7])

$s^α=sup{η: there exists anα-strong sequence of cardinality η}.$

Then by Theorem 2.5 we have

(GCH). 𝔩κŝκ+.

3 Some further results

Definition 3.1

Let Q be a countable dense subset of the interval [0, 1]. Then X ⊆ [0, 1] is called concentrated on Q if every open set of [0, 1] containing Q, contains all but countably many elements of X.

Assuming CH, Lemma 2.3 and Proposition 3.4 in [1] and Theorem 2.5 we have the following proposition.

Proposition 3.2

(CH). The following are equivalent

1. there exists a1-strong sequence (Hξ)ξ < 20, with |Hξ| < 20, for any ξ < 20,

2. 𝔩 = ℵ1,

3. there exists a K-Lusin set of cardinality 𝔠,

4. there exists a concentrated set of cardinality 𝔠,

5. there exists a BK-matrix.

The following fact is obvious, (see [1], Lemma 2.2).

Fact 3.3

Every Lusin set is a K-Lusin set.

The following corollary is immediate.

Corollary 3.4

If there exists a Lusin set of cardinality 𝔠 (𝔠 - regular) then there is an1-strong sequence (Hξ)ξ < 20 with |Hξ| < 20 for any ξ < 20.

Following Lemma 8.26 in [8] we obtain a Lusin set of size κ by adding κ many Cohen reals. By Corollary 2.6 we have that it is consistent with ZFC that there exists ℵ1-strong sequence from Proposition 3.2(a). On the other hand we know that 𝔩 = ℵ1 implies 𝔟 = ℵ1 and 𝔡 = 𝔠, ([1], Lemma 2.4). It is consistent with ZFC that 𝔟 > ℵ1 or 𝔡 < 𝔠, (cf. [8]). Thus there are no such ℵ1-strong sequences. Summing up we have the following result ([1], Theorem 2.6.).

Theorem 3.5

The existence of1-strong sequence (Hξ)ξ < 20 with |Hξ| < 20 for any ξ < 20 is independent of ZFC+¬CH.

In the proof of Proposition 3.2 in [1] the authors used the ω2-iteration of Miller forcing with countable support over V, a model of ZFC + CH showing that the following property holds. “For all fωωV[Gω2] the set {ı : mıf} is countable”, where Gω2 = 〈mı : ı < ω2〉 is the corresponding sequence of Miller reals. Using Proposition 3.2 and the previous considerations we obtain the following proposition.

Proposition 3.6

It is consistent with ZFC that there exists1-strong sequence (Hξ)ξ < 20 with |Hξ| < 20 for any ξ < 20.

Before Theorem 3.5 there was mentioned that 𝔩 = ℵ1 (so the existence of ℵ1-strong sequence (Hξ)ξ < 20 with |Hξ| < 20 for any ξ < 20 by Proposition 3.2) implies that 𝔟 = ℵ1 and 𝔡 = 𝔠. We can construct a model of ZFC in which although 𝔟 = ℵ1 and 𝔡 = 𝔠 there is no such an ℵ1-strong sequence (Hξ)ξ < 20 with |Hξ| < 20 for any ξ < 20, (compare the proof of Proposition 3.3 in [1]).

Proposition 3.7

It is consistent with ZFC that 𝔟 = ℵ1 and 𝔡 = 𝔠 but there is no1-strong sequence (Hξ)ξ<𝔠 with |Hξ| < 20 for any ξ < 𝔠.

Proof

Let us start with a model M in which 𝔠 = ℵ2 and in which MA holds. Let G = 〈cβ : β < ω1〉 be a generic sequence of Cohen reals of length ω1. The first part of the proposition is fulfilled but there is no ℵ1-strong sequences of required property. Indeed. Let (Hξ)ξ<ℵ2, |Hξ| < ℵ2 be an ℵ1-strong sequence of length ℵ2. Consider X = ⋃ξ<ℵ2 Hξ. Let Gα = 〈cβ : β < α〉 for some α-countable. Let X′ ⊆ X be of cardinality ℵ2 and X′ ⊆ M[Gα]. Now M[Gα] = M[c] for some Cohen real c and M[c]⊩ MA(σ − centered) which implies that 𝔭 = 𝔠. We know that 𝔭 ≤ 𝔟, (see e.g. [8]), thus M[c]⊩ 𝔟 = ℵ2. It means that there exists a function such that |{fX′ : f}| = ℵ2. We obtain a contradiction with the definition of an ℵ1-strong sequence. □

4 On the existence of generalized BK-matrix and generalized strong sequences

A BK-matrix proposed in [2], (see also [1]), is a special kind of a generalized independent family, well-known in the literature (see [4,5,6]).

Definition 4.1

Let 𝓘 = {{ $\begin{array}{}{I}_{\alpha }^{\beta }\end{array}$ : β < λα} : α < τ} be a family of partitions of infinite set S with each λα ≥ 2 and let κ, λ, θ be cardinals. If for any J ∈ [τ]<θ and for any fΠαJ λα the intersection ⋂{ $\begin{array}{}{I}_{\alpha }^{f\left(\alpha \right)}\end{array}$ : αJ} has cardinality at least μ, then 𝓘 is called (θ, μ)-generalized independent family on S. Moreover, if λα = λ for all α < τ, then 𝓘 is called a (θ, μ, λ)-generalized independent family on S.

Using Definition 4.1 we conclude that BK-matrix is (ω1, 1, ω)- generalized independent family. Using the proof of Proposition 1.1. in [1] we obtain similar result for a generalized BK-matrix (i.e. BK-matrix considered in κκ instead of [0, 1], where κ is an infinite cardinal). Moreover, using Theorem 2.5 we obtain that

Proposition 4.2

(GCH). Let κ be an infinite cardinal. On each set of cardinality κ+ the following are equivalent:

1. there exists (κ+, 1, κ)-generalized independent family of cardinalit 2κ,

2. there exists κ+-strong sequence of cardinality κ+.

5 Further applications

In [7] there are some results concerning the existence of ω-strong sequences and their dependence on a precalibre. We generalize those results as follows. Let α and κ be cardinals.

Definition 5.1

A cardinal η is a precalibre for Xκκ if η is infinite and every set A ∈ [X]η has an α-directed set in the sense of relationof cardinality η, with α ≤ 2κ.

Proposition 5.2

Let κ be an infinite cardinal. If a regular cardinal η is not a precalibre for Xκκ, then there exists a κ+-strong sequence of length η.

Proof

If η is not a precalibre, then there exists A ∈ [X]η of cardinality η in which each κ+-directed subset has cardinality less than η.

Let f0A be an arbitrary function and let A0A be a maximal κ+-directed subset such that f0A0. Let A0 be the first element of an κ+-strong sequence.

Assume that there has been defined the sequence of functions {fξ:ξ < ψ < η} with fξfψ for any ξ < ψ such that for each fξ there exists a maximal, κ+-directed set AξA such that fξAξ and (Aξ)ξ<ψ forms the κ+-strong sequence. Since |Aξ| < η and η is regular we have that A ∖ ⋃ξ<ψ Aξ is non-empty. Hence there exists fζA ∖ ⋃ξ<ψ Aξ such that fζfξ for all ξ < ζ and AζA ∖ ⋃ξ<ψ Aξ is maximal, κ+-directed such that fζAζ. Let Aζ be the next element of the κ+-strong sequence. □

Using Proposition 5.2, Theorem 2.5, Proposition 4.2 and Lemma 2.2 we immediately obtain the following corollary

Corollary 5.3

Let κ be an infinite cardinal. If a regular cardinal κ is not a precalibre then

1. 𝔩κ = 2κ,

2. there exists (κ+, 1, κ)-generalized independent family of cardinality 2κ,

3. there exists a generalized K-Lusin set of cardinality 2κ.

References

• [1]

Bartoszyński T., and Halbeisen L., On a theorem of Banach and Kuratowski and K-Lusin sets, Rocky Mt. J. Math., 2003, 33, 1223-1231

• [2]

Banach S., Kuratowski K., Sur une généralisation du probléme de la mesure, Fund. Math., 1929, 14, 127-131

• [3]

Efimov B. A., Diaditcheskie bikompakty, (in Russian), Trudy Mosk. Matem. O-va, 1965, 14, 211-247 Google Scholar

• [4]

Elser S. O., Density of κ-Box Products and the existenxce of generalized independent families, App. Gen. Top., 2011, 12(2), 221-225 Google Scholar

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Hu W., Generalized independent families and dense sets of Box-Product spaces, App. Gen. Top., 2006, 7(2), 203-209

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Jureczko J., κ-strong sequences and the existence of generalized independent families, Open. Math., 2017, 15, 1277-1282

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Jureczko J., On inequalities among some cardinal invariants, Math. Aeterna, 2016, 6(1), 87 - 98 Google Scholar

• [8]

Bartoszyński T., Judah T., On the structure of the real line, 1995, A. K. Peters, Wellesley, Massachusetts

Accepted: 2018-04-19

Published Online: 2018-07-04

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 724–729, ISSN (Online) 2391-5455,

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