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

Joanna Jureczko

doi:10.1515/math-2018-0066

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

Joanna Jureczko

Published/Copyright: July 4, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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.

Keywords: GCH; Consistency results; BK-matrix; Lusin set; Strong sequences

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 2^ℵ₀ 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 matrixAki ⊆ [0, 1] (wherei, k ∈ ω) such that

for eachi ∈ ω, [0, 1] = ⋃_k∈ωAki,
for eachi ∈ ω, ifk ≠ k′ thenAki ∩ Ak′i = ∅,
for every sequencek₀, k₁, …, k_i, … ofω 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, g ∈ ^ωω, f ⪯ g iff |{n ∈ ω : 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 κ ≥ ω be a regular cardinal. In the set ^κκ we consider the following relation, if f, g ∈ ^κκ then g ⪯ f iff |{α ∈ κ : g(α) > f(α)}| < κ.

A set K ⊆ ^κκ is said to be closed iff for all f ∈ K and for all g ∈ ^κκ if f ⪯ g then g ∈ K. A closed set K ⊆ ^κκ is κ-compact iff there is a function f ∈ ^κκ such that K ⊆ {g ∈ ^κκ : g ⪯ f}.

Definition 2.1

A setXof size 2^κis ageneralized K-Lusin set if |X ∩ K| < 2^κfor everyκ-compact setK ⊆ ^κκ.

Now let 𝓕 ⊆ ^κκ then

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

Assuming GCH we obtain 𝔩_κ = 2^κ. 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 {f_α ∈ ^κκ : α < κ⁺} such that

f_β ⪯ f_α for β < α < κ⁺
g_α ⪯ f_α for α < κ⁺

by transfinite recursion.

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

Let α be a successor. Take g_α ∈ ^κκ and suppose that for each f_α ∈ ^κκ ∖ {f_β ∈ ^κκ : β < α} at least one of the conditions (i) or (ii) does not hold.

If (ii) does not hold then |{f ∈ ^κκ : f ⪯ g_β}| = 2^κ for β < α . A contradiction. The argumentation for proving that (i) holds is similar.

If α is limit, then we take f_α = ⋃_β<αf_β. □

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

Lemma 2.3

(GCH). The following are equivalent

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

Now we define a generalized strong sequence, (see also [6]). Let α be a cardinal number. We say that A ⊆ ^κκ is anα-directed set iff for each B ⊂ A of cardinality less than α there exists f ∈ ^κκ such that g ⪯ f for all g ∈ B.

Definition 2.4

Letαandηbe cardinals. A sequence (H_ϕ)_ϕ<η, whereH_ϕ ⊂ X, is called anα-strong sequence if:

H_ϕisα-directed for allϕ < η,
H_ψ ∪ H_ϕis notα-directed for allϕ < ψ < η.

The following result is true.

Theorem 2.5

(GCH). The following are equivalent

𝔩_κ = 2^κ
there exists aκ⁺-strong sequence (H_ξ)_ξ<κ⁺with |H_ξ| < κ⁺for allξ < κ⁺.

Proof

By Lemma 2.2 there exists a family 𝓕 ⊆ ^κκ of cardinality κ⁺ with λ_κ(𝓕) = 2^κ. Choose f₀ ∈ ^κκ arbitrarily. Let H₀ = {f ∈ 𝓕 : f ⪯ f₀}. Hence |H₀| < 2^κ. Let H₀ be the first element of a κ⁺-strong sequence.

Assume that the κ⁺-strong sequence (H_ξ)_ξ<ψ for ψ < κ⁺ such that

Hξ={f∈F∖⋃η<ξHη:f⪯fξ},

where f_ξ ∈ ^κκ ∖ {f_η : η < ξ}, f_ξ ⊥ f_η, (f_ξ ⊥ f_η means f_η ⪯ f_ξ and f_ξ ⪯ f_η for η < ξ) has been defined.

Since ψ < κ⁺, |𝓕| = κ⁺ and κ⁺ is regular, we have that there exists f ∈ ^κκ ∖ {f_ξ : ξ < κ⁺} such that f ⊥ f_ξ for any ξ < ψ. Name such f by f_ψ. Let H_ψ = {f ∈ 𝓕 ∖ ⋃_{ξ < ψ}H_ξ : f ⪯ f_ψ. Obviously |H_ψ| < κ⁺, (because of our assumptions). By the construction H_ξ ∪ H_ψ, ξ < ψ is not κ⁺-directed, (because f_ψ ⊥ f_ξ for any ξ < ψ).

Now let (H_ξ)_ξ<κ⁺ be a κ⁺-strong sequence with H_ξ ⊆ ^κκ and |H_ξ| < κ⁺ for each ξ < κ⁺. According to Definition 2.4 we have that for all ξ < ψ there exist h_ξ ∈ H_ξ, h_ψ ∈ H_ψ such that h_ξ ⊥ h_ψ. For any ψ < κ⁺ let

Hψ′={hψ∈Hψ:ξ<ψ⇒∃hξ∈Hξhξ⊥hψ}.

Since H_ψ, ψ < κ⁺ is κ⁺-directed, there exists h̄_ψ ∈ ^κκ such that h ⪯ h̄_ψ, for all h_ψ ∈ Hψ′ . Thus we can describe H_ψ as follows

Hψ={f∈κκ:f⪯h¯ψ}.

Let 𝓕 = ⋃{H_ξ : ξ < κ⁺}. For any g ∈ ^κκ consider {f ∈ 𝓕 : f ⪯ g}. Suppose that there exists g_o ∈ ^κκ such that |{f ∈ 𝓕 : f ⪯ g₀}| = κ⁺. But by definition of 𝓕, there exists ξ₀ such that {f ∈ 𝓕 : f ⪯ g₀} ⊆ H_ξ₀. Hence |H_ξ₀| = κ⁺. A contradiction. □

The next corollary follows from Theorem 2.5 and Lemma 2.3.

Corollary 2.6

(GCH). The following are equivalent

there exists a generalized K-Lusin set of cardinalityκ⁺
there exists anκ⁺-strong sequence (H_ξ)_ξ<κ⁺, H_ξ ⊆ ^κκwith |H_ξ| < κ⁺for anyξ < 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

Corollary 2.7

(GCH). 𝔩_κ ≤ ŝ_κ⁺.

3 Some further results

Definition 3.1

LetQbe a countable dense subset of the interval [0, 1]. ThenX ⊆ [0, 1] is called concentrated onQif every open set of [0, 1] containingQ, contains all but countably many elements ofX.

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

there exists a ℵ₁-strong sequence (H_ξ)_ξ < 2^ℵ₀, with |H_ξ| < 2^ℵ₀, for any ξ < 2^ℵ₀,
𝔩 = ℵ₁,
there exists a K-Lusin set of cardinality 𝔠,
there exists a concentrated set of cardinality 𝔠,
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 an ℵ₁-strong sequence (H_ξ)_ξ < 2^ℵ₀with |H_ξ| < 2^ℵ₀for anyξ < 2^ℵ₀.

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 of ℵ₁-strong sequence (H_ξ)_ξ < 2^ℵ₀with |H_ξ| < 2^ℵ₀for anyξ < 2^ℵ₀is 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 f ∈ ^ωω ∩ V[G_ω₂] the set {ı : m_ı ⪯ f} is countable”, where G_ω₂ = 〈m_ı : 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 exists ℵ₁-strong sequence (H_ξ)_ξ < 2^ℵ₀with |H_ξ| < 2^ℵ₀for anyξ < 2^ℵ₀.

Before Theorem 3.5 there was mentioned that 𝔩 = ℵ₁ (so the existence of ℵ₁-strong sequence (H_ξ)_ξ < 2^ℵ₀ with |H_ξ| < 2^ℵ₀ for any ξ < 2^ℵ₀ 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 (H_ξ)_ξ < 2^ℵ₀ with |H_ξ| < 2^ℵ₀ for any ξ < 2^ℵ₀, (compare the proof of Proposition 3.3 in [1]).

Proposition 3.7

It is consistent with ZFC that 𝔟 = ℵ₁and 𝔡 = 𝔠 but there is no ℵ₁-strong sequence (H_ξ)_ξ<𝔠with |H_ξ| < 2^ℵ₀for anyξ < 𝔠.

Proof

Let us start with a model M in which 𝔠 = ℵ₂ and in which MA holds. Let G = 〈c_β : 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 (H_ξ)_ξ<ℵ₂, |H_ξ| < ℵ₂ be an ℵ₁-strong sequence of length ℵ₂. Consider X = ⋃_ξ<ℵ₂H_ξ. Let G_α = 〈c_β : β < α〉 for some α-countable. Let X′ ⊆ X be of cardinality ℵ₂ and X′ ⊆ M[G_α]. Now M[G_α] = 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 ḡ such that |{f ∈ X′ : f ⪯ ḡ}| = 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 = {{ Iαβ : β < λ_α} : α < τ} be a family of partitions of infinite setSwith each λ_α ≥ 2 and letκ, λ, θbe cardinals. If for anyJ ∈ [τ]^<θand for anyf ∈ Π_α∈J λ_αthe intersection ⋂{ Iαf(α) : α ∈ J} has cardinality at leastμ, then 𝓘 is called (θ, μ)-generalized independent family onS. Moreover, if λ_α = λ for allα < τ, then 𝓘 is called a (θ, μ, λ)-generalized independent family onS.

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). Letκbe an infinite cardinal. On each set of cardinalityκ⁺the following are equivalent:

there exists (κ⁺, 1, κ)-generalized independent family of cardinalit 2^κ,
there existsκ⁺-strong sequence of cardinalityκ⁺.

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 cardinalηis a precalibre forX ⊆ ^κκifηis infinite and every setA ∈ [X]^ηhas anα-directed set in the sense of relation ⪯ of cardinalityη, withα ≤ 2^κ.

Proposition 5.2

Letκbe an infinite cardinal. If a regular cardinalηis not a precalibre forX ⊆ ^κκ, then there exists aκ⁺-strong sequence of lengthη.

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 f₀ ∈ A be an arbitrary function and let A₀ ⊆ A be a maximal κ⁺-directed subset such that f₀ ∈ A₀. Let A₀ be the first element of an κ⁺-strong sequence.

Assume that there has been defined the sequence of functions {f_ξ:ξ < ψ < η} with f_ξ ⊥ f_ψ for any ξ < ψ such that for each f_ξ there exists a maximal, κ⁺-directed set A_ξ ⊆ A such that f_ξ ∈ A_ξ and (A_ξ)_ξ<ψ forms the κ⁺-strong sequence. Since |A_ξ| < η and η is regular we have that A ∖ ⋃_ξ<ψA_ξ is non-empty. Hence there exists f_ζ ∈ A ∖ ⋃_ξ<ψA_ξ such that f_ζ ⊥ f_ξ for all ξ < ζ and A_ζ ⊂ A ∖ ⋃_ξ<ψA_ξ is maximal, κ⁺-directed such that f_ζ ∈ A_ζ. Let A_ζ 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

Letκbe an infinite cardinal. If a regular cardinalκis not a precalibre then

𝔩_κ = 2^κ,
there exists (κ⁺, 1, κ)-generalized independent family of cardinality 2^κ,
there exists a generalized K-Lusin set of cardinality 2^κ.

References

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Received: 2017-08-08

Accepted: 2018-04-19

Published Online: 2018-07-04

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Articles in the same Issue

https://doi.org/10.1515/math-2018-0066

Keywords for this article

GCH; Consistency results; BK-matrix; Lusin set; Strong sequences

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