κ-strong sequences and the existence of generalized independent families

Joanna Jureczko

doi:10.1515/math-2017-0108

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κ-strong sequences and the existence of generalized independent families

Joanna Jureczko

Published/Copyright: November 10, 2017

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$Open Mathematics$

From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper we will show some relations between generalized versions of strong sequences introduced by Efimov in 1965 and independent families. We also show some inequalities between cardinal invariants associated with these both notions.

Keywords: Strong sequences; Generalized independent families; Cardinal invariants

MSC 2010: 03E20; 54A05; 54A25

1 Introduction

The strong sequences method was introduced by Efimov in 1965 [1] as a useful tool for proving some famous theorems in dyadic spaces, (i.e. continuous images of Cantor cube). Among others Efimov proved that strong sequences do not exist in the subbase of Cantor cube. The problem of the existence of strong sequences in the other spaces was raised in 90’s of the last century (see [2] or [3] for more historical details).

The problem of the existence of independent families was raised among others by Fichtenholz and Kantorovitch [4] and Hausdorff [5]. There was shown that for each cardinal κ there exists an independent family of size 2^κ. Engelking and Karłowicz introduced in [6] a generalization of the definition of this notion by considering partitions of a given set instead of a pair of sets. However, the name “generalized independent family” was used for the first time by Hu in [7]. In [8] Elser proved among others that under some assumptions there exists a generalized independent family (see Corollary 2.6.), using the result obtained by Hu (see Theorem 2.4 in [7]).

The aim of this paper is to show relations between generalized independent families and generalized strong sequences. There will be also shown relations between cardinal invariants concerning both mentioned notions.

The notation used in the paper is standard for the field and can be found in e. g. [9] and [10].

2 Generalized strong sequences

Let (X, r) be a set with an arbitrary relation r and let κ be a cardinal. By P(X) we denote the power set of X, i. e. the family of all subsets of X. We say that a set A ⊂ X has a bound iff there exists b ∈ X such that for all a ∈ A we have (a, b) ∈ r. (If elements a, b ∈ X have not a bound we say that they are incompatible). We say that a set A ⊂ X is κ–directed iff each subset of A of cardinality less than κ has a bound.

Definition 2.1

Let (X, r) be a set with a relation r. A sequence (H_ϕ)_{ϕ < α}, where H_ϕ ⊂ X, is called a κ-strong sequence if:

1^o H_ϕ is κ-directed for all ϕ < α

2^o H_ψ ∪ H_ϕ is not κ-directed for all ϕ < ψ < α.

Let β and τ be cardinals. By β≪τ we denote: τ is strongly β-inaccessible, i. e. β < τ and α^λ < τ whenever α < β, λ < τ.

The following theorem is the generalization of Theorem 3 in [3]).

Theorem 2.2

(on κ-strong sequences). Let β, κ, μ, τ be cardinals such that ω ≤ β≪τ, μ < β, κ < τ and β, τ be regular. Let X be a set of cardinality τ. If there exists a κ-strong sequence {H_α ⊂ X:α < τ} with |H_α| ≤ 2^μ for all α < τ, then there exists a κ-strong sequence {T_α:α < β} with |T_α| < κ for all α < β.

Proof

Consider a κ-strong sequence {H_α:α < τ} as was done in the theorem. We define families {P_{α_γ}} ∈ 𝓟(X):γ < β} with the following properties:

{α_γ:γ < β} is an increasing subsequence of τ;
Pαγ={Tαγg∈[Hαγ]<κ:|fαγ,h−1(Tαγg)|=τ, for some h ∈ ^δτ, g ∈ γ_τ, δ < γ};
fαγ,h:Aαδh∖{αγ}→[Hαγ]<κ is a function given by the formula fαγ,h(ξ)=Tαγg,h∈δτ,g∈γτ,δ<γ, with the property Tαγg∪Hα and Tαγg∪Tαδh are not κ-directed for α_δ < α_γ < α, where Aαδh=fαδ,h−1(Tαδg) and |Aαδh|=τ for δ<γ,Tαδg∈Pαδ.

We proceed by transfinite recursion. Take α₀ < τ and H_α₀. (Without the loss of generality we can assume that α₀ = 0). By Definition 1 for each α > α₀ there exists T ∈[H_α₀]^{< κ} such that T ∪ H_α is not κ-directed. Consider a function

fα0:τ∖{α0}→[Hα0]<κ

given by the formula fα0(ξ)=Tα0ξ, where Tα0ξ∈[Hα0]<κ is such that Tα0ξ∪Hα is not κ–directed for any α₀ < α. Since |H_α₀| ≤ 2^μ we have |[H_α₀]^{< κ}| ≤ (2^μ)^{< ω} = 2^μ. Then the function f_α₀ determines a partition of τ ∖ {α₀} into at most 2^μ < τ elements. Since τ is regular the family

Pα0={Tα0ξ∈[Hα0]<κ:|fα0−1(Tα0ξ)|=τ, for some ξ<τ}

is non-empty.

Next for η < β we define P_{α_η+1}. By previous steps for η < β we have constructed P_{α_η} = { Tαηg ∈ [H_{α_η}]^{< κ}:| fαη,h−1(Tαηg) | = τ, for some h ∈ ^δτ, g ∈ ^ητ, δ < η}. For each Tαηg ∈ P_{α_η} let Aαηg = fαη,h−1(Tαηg) . For each such a set Aαηg consider a function

fαη+1,g:Aαηg∖{αη+1}→[Hαη+1]<κ

given by the formula fαη+1,g(ξ)=Tαη+1g⌢ξ for Tαη+1g⌢ξ∈[Hαη]<κ such that Tαη+1g⌢ξ∪Hα and Tαη+1g⌢ξ∪Tαηg are not κ-directed whenever α_δ < α_η < α. Since |H_{α_η+1}| ≤ 2^μ we have |[H_{α_η+1}]^{< κ}| ≤ (2^μ)^{< κ} = 2^μ, the function f_{α_η+1,g} determines a partition of Aαηg ∖ {α_η+1} into at most 2^μ < τ elements. Since τ is regular the family

Pαη+1={Tαη+1g⌢ξ∈[Hαη+1]<κ:|fαη+1,g−1(Tαη+1g⌢ξ)|=τ, for some g⌢ξ∈η+1τ}

is non-empty and fulfills (ii)-(iii).

Now we assume that η is limit and for all δ < η the families P_{α_δ} = { Tαδg ∈[H_{α_δ}]^{< κ}:| fαδ,h−1(Tαδg) | = τ, for some h ∈ ^λτ, g ∈ ^δτ, λ < δ} fulfilling (ii)-(iii) have been defined. We set

Bαη={⋂δ<η⋃g∈δτTαδg:Tαδg∈Pαδ}

and

Bαη′={T∈Bαη:T≠∅}.

Obviously |Bαη′|<_2μ. By previous steps we have |fαδ,h−1(Tαδg)|=τ and Aαδh=fαδ,h−1(Tαδg) for all Tαδg∈Pαδ and h ∈ ^λτ, g ∈ ^δτ, λ < δ < τ. Let

Fαδ={h∈δτ:|fαδ,h−1(Tαδg)|=τ}.

Take Aαη=⋃h∈FαδAαδh. Consider a function

fαη:Aαη∖{αη}→[Bαη′]<κ

given by the formula fαη(ξ)=Tαηξ such that Tαηξ∪Hα and Tαηξ∪Tαδh are not κ-directed for all h ∈ F_{α_δ}, α_δ < α_η < α. Then the family Pαη={Tαηξ∈[Bαη]<κ:|fαη−1(Tαηξ)|=τ for some ξ < τ} fulfills (ii)-(iii).

Thus we have constructed at least one κ-strong sequence of the form

{Tαγg∈Pαγ:γ<β,g∈γτ}.

Suppose now that at least one of defined above κ-strong sequences has length ζ > β. Each set Tαγg determines a set Aαγg ⊂ τ of cardinality τ, i. e. there is defined a function f_{α_γ,g} as in (iii). Let v = sup{|P_{α_γ}|:γ < ζ}. Then there would exist v^ζ > τ pairwise disjoint sets Aαγg of cardinality τ. A contradiction. □

Corollary 2.3

Let β, κ, τ be cardinals such that ω ≤ β ≪ τ, κ < τ and β, τ be regular. Let X be a set of cardinality τ. Then either X contains a κ-directed subset of cardinality τ or there exists a family of cardinality β consisting of subsets of [X]^{< κ} with the property: for each A, B ∈[X]^{< κ} the set A ∪ B is not κ-directed.

Proof

Without the loss of generality we can assume that X ⊆ τ. Suppose that each κ-directed subset of X has cardinality less than τ. We will construct a κ-strong sequence {H_α:α < τ} via transfinite recursion.

Assume that for α < τ the κ-strong sequence {H_η ⊂ X ∖ ⋃_{γ < η} H_γ:η < α} has been defined. Since |H_η| < τ then |⋃_{η < α}H_η| < τ and τ is regular we have |X ∖ ⋃_{η < α}H_η| = τ. Now we will construct H_α.

Let α be successor. Let H_α ⊂ X ∖ ⋃_{η < α}H_η be a maximal κ-directed set.

Let α be limit. Let x = min(X ∖ ⋃_{η < α}H_η). Let H_α = ⋃_{η < α}H_η ∪{x}.

Let H_α be the next element of the κ-strong sequence. By Theorem 2.2 there exists a κ-strong sequence {T_α:α < β} such that |T_α| < κ for all α < β. If T_α are not pairwise disjoint then we take the family consisting of sets Tα′ such that T0′ T₀ and Tα′ = T_α ∖ ⋃_{β < α}T_β for α -successor and Tα′ = sup(⋃_{β < α} Tβ′ ) for α -limit. This completes the proof. □

The next corollary follows immediately from Corollary 2.3.

Corollary 2.4

Let β, κ, τ be cardinals such that ω ≤ β ≪ τ, κ < τ and β, τ be regular. Let X be a set of cardinality τ. Then either X contains a κ-directed subset of cardinality τ or there exists a subset of X of cardinality β consisting of pairwise incompatible elements.

The next result in this paragraph will be the special case of previous ones for (P(X), ⊆).

A family 𝓐 ⊂ P(X) is closed under taking κ -intersections i.e. for all 𝓐′ ⊂ 𝓐 such that |𝓐′| < κ we have ∩ 𝓐′ ∈ 𝓐. Let κ, τ be cardinals with κ < τ. A family of sets 𝓐 ⊂ P(X), with |𝓐| ≥ τ, is called a κ-vaulted family iff for each subfamily 𝓑 ⊂ 𝓐 of cardinality less than κ we have ∩ 𝓑 ≠ ∅.

Theorem 2.5

Let β, κ, τ be cardinals such that ω ≤ β≪τ, κ < τ and β, τ be regular Let X be a topological space of cardinality τ. Let 𝓐 ⊂ 𝓟(X) be a family of sets with |𝓐| = τ closed under taking κ-intersections. Then 𝓐 contains a κ-vaulted family of cardinality τ or 𝓐 contains a subfamily of cardinality β which consists of pairwise disjoint sets.

Proof

Let 𝓐 = {A_γ:γ < τ} be a family as is required in the theorem. Define a partial ordered set 𝓟 = {γ < τ:A_γ ∈ 𝓐} with the following relation.

(γ,β)∈r⇔Aγ⊆Aβ.

If γ, β are incompatible, then A_γ ∩ A_β = ∅. By Corollary 2.4 the proof is complete. □

3 Generalized independent families

In [7] the following definition was introduced

Definition 3.1

([7]). Let 𝓘 = {{ Iαβ : β < λ_α}:α < τ} be a family of partitions of an infinite set S with each λ_α ≥ 2 and let κ, λ, θ be cardinals. If for any J ∈[τ]^<θ and for any f ∈Π_{α ∈ J}λ_α the intersection ⋂{ Iαf(α) : α ∈ J} has cardinality at least κ, then 𝓘 is called a(θ, κ)-generalized independent family on S. Moreover, if λ_α = λ for all α < τ, then 𝓘 is called a(θ, κ, λ)-generalized independent family on S.

Notice that an independent family considered in [6] is (ω, 1, |S|)-independent family on a set S. Moreover in [8] and [7] there are shown some results concerning cardinality of generalized independent families. In Theorem 3.2 we show the relation between the existence of generalized independent families and κ-strong sequences.

Let 𝓐 ⊂ P(X). Then

c(A)=sup{|B|:B is a subfamily of pairwise disjoint sets of A}.

Theorem 3.2

Let β, κ, τ be cardinals such that ω ≤ β≪τ, κ < τ and β, τ be regular Let X be a topological space of cardinality τ. Let 𝓐 ⊂ 𝓟(X) be a family of cardinality τ closed under taking κ-intersections and such that c(𝓐) < β. Then there exists a(κ, 1)-generalized independent family of cardinality τ.

Proof

Consider a family Part = {P^α : |P^α| = λ_α, α < ξ} of all partitions of τ. Since λ_α ≤ τ for all α < ξ, |Part|≥τ. By Theorem 2.5 there exists a κ-vaulted family 𝓐 ⊂ Part of cardinality τ. We will construct a (κ, 1)-generalized independent family via transfinite recursion.

Assume that for η < τ the (κ, 1)-generalized independent family 𝓘 = {P^α} ∈ 𝓐\ {P^γ}:γ < α}:α < η} has been defined. Clearly, 𝓘 has the property that for any J ∈[η]^{< κ} and f ∈Π_α∈J λ_α the intersection ⋂{ Iαf(α) :α ∈ J} is non-empty, where Iαf(α) is a f(α)-element of the partition P^α. Since η < τ and |𝓐| = τ, 𝓐\ {P^α:α < η}≠∅. Hence we can continue our construction. Let

I={I:I=⋂{Iαf(α):α∈J} for some J∈[η]<κ and some f∈Πα∈Jλα}.

Observe that ∪𝓘 = τ. If not, then there exists δ ∈τ\ ∪𝓘. It would mean that there is no J ∈[η]^<κ and f ∈Π_α∈Jλ_α such that δ ∈⋂{ Iαf(α) : α ∈ J}. Then δ∉P^α for some P^α ∈ 𝓘. A contradiction because P^α ∈ Part.

Let η be a successor. Let P^η ∈ 𝓐\ {P^α : α < η} be a partition with the property that for all I ∈ 𝓘 there exists I_η ∈ P^η such that I ⊂ I_η. For a set J ∈[τ]^<κ and f ∈Π_α∈J λ_α choose { Iαf(α) : α ∈ J}. Since the family 𝓐 is κ-vaulted, ⋂{ Iαf(α) : α ∈ J}≠∅.

If η is the limit, then we take P^η = ⋂_α<η P^α.

Thus the (κ, 1)-generalized independent family {P^α = { Iαβ : β < λ_α}:α < τ} has been defined.□

In [6] it is proved the result related to the density of product of topological spaces (Theorem 8) using the theorem on the existence of independent families (Theorem 3). Following [8], we give the definition of κ-box product.

Let κ, μ be cardinals with ℵ₀ ≤ κ ≤ μ and {X_i}_i∈μ be a family of topological spaces. Then ◻i∈μκXi denotes the κ-box product which is induced on the full Cartesian product Π_i∈μ X_i by the canonical base

B={⋂i∈Ipri−1(Ui):I∈P<κ(μ) and Ui is open in Xi},

where P_<κ(μ) : = {I ⊂μ:|I| < κ}.

The next two corollaries follow from Theorem 2.5 and Corollary 2.3 and Theorem 4.3 in [7].

Corollary 3.3

Let β, κ, τ be cardinals such that ω ≤ β≪τ, κ < τ and β, τ be regular Let X be a topological space of cardinality τ. Let 𝓐 ⊂ 𝓟(X) be a family of cardinality τ closed under taking κ-intersections and such that c(𝓐) < β. Let {X_α}_α<τ be a family of topological spaces such that d(X_α) ≤ λ_α for all α < τ. Then d(◻α∈κτ(Xα))≤|S|.

A topological space X is irresolvable if X does not have disjoint dense subsets.

Let λ be a cardinal. An ideal 𝓘 ⊂ P(X) is called λ-complete if ⋃_α<λ A_α ∈ 𝓘 for α < λ and A_α ∈ 𝓘.

In [7], one can find the following lemma

Lemma 3.4

Suppose (X, 𝓣) is an open-hereditarely irresolvable space and 𝓣 is a P_θ-topology for some regular cardinal θ. Let 𝓝 denote the ideal of nowhere dense subsets, and let λ be the smallest cardinal such that 𝓝 is not λ-complete. Then for any γ < γ⁺ < λ and η < θ, 𝓝 is (γ^η)⁺-complete.

Corollary 3.5

Let β, κ, θ, τ be cardinals such that ω ≤ β≪τ, κ < τ and θ < τ with β, τ, θ are regular Let X be a topological space of cardinality τ. Let 𝓐 ⊂ 𝓟(X) be a family of cardinality τ closed under taking κ-intersections and such that c(𝓐) < β. Suppose that 𝓘 is a family of partitions of a set S with each λ_α < θ. Let 𝓝 be the ideal of nowhere dense set of the simple topology induced by 𝓘 and let λ be the smallest cardinal such that 𝓘 is not λ- complete. Then

there is a nonempty open set U of the simple topology such that U with the subspace topology satisfies all conditions in Lemma 3.4 and the ideal 𝓘_U of nowhere dense set of U is λ-saturated and
2^<θ = θ.

We finish this paper by showing relations between cardinal invariants associated with both considered notions. Let κ be a cardinal and let X be an infinite set of the cardinality κ. Accept the following notations:

s^κ=sup{α: there exists a κ-strong sequence in X of length α}.i(κ,1)=min{α: there is no (κ,1)− generalized independent family on X of length α}.

Theorem 3.6

Let κ, τ be cardinals such that κ < τ and τ-regular and there exists a regular cardinal β such that ω ≤ β≪τ. Let X be a topological space of cardinality τ. Then ŝ_κ ≤ i(κ, 1).

Proof

Without the loss of generality we can assume that X ⊆τ. Let 𝓘 be a maximal (κ, 1)-generalized independent family. If |𝓘| = τ and 𝓘 contains a κ-vaulted family of cardinality τ then the theorem is complete. Suppose that each κ-vaulted family has cardinality less than τ. (Then by Theorem 2.5 there are only β pairwise disjoint sets in 𝓘 for ω ≤ β≪τ, β, τ-regular). By transfinite recursion we will construct a κ-strong sequence {H_α:α < τ}. Assume that for γ < α < τ the κ-strong sequence {H_γ : γ < α} such that H_γ ⊂ X\ ⋃_{η < γ}H_η has been defined. Since |H_γ| < τ, |⋃_γ<αH_γ|<τ and τ is regular we have |X \ ⋃_{γ< α}H_γ| = τ.

Let α be a sucessor. Hence there exists a maximal κ-directed set H_α ⊂ X\ ⋃_{γ< α} H_γ such that H_α ∪ H_γ is not κ-directed for all γ < α.

If α is limit, then H_α = ⋃_{γ< α}H_γ ∪{x}, where x = min(X \ ⋃_{γ< α} H_γ). Obviously, H_α ∪ H_γ is not κ-directed for all γ < α. Let H_α be the next element of the κ-strong sequence. The proof is complete.□

The easy consequence of Theorem 3.6 and Theorem 3.2. in [7] is

Corollary 3.7

Let κ, τ be cardinals such that κ < τ and τ -regular and there exists a regular cardinal β such that ω ≤ β≪τ. Let X be a topological space of cardinality τ. Then the following statements are equivalent:

ŝ_κ ≤ i(κ, 1)
d(◻κs^κ(Xα))≤|X| holds for any family of topological spaces {X_α}_{α<ŝ_κ}, with each d(X_α) ≤ λ for some λ < τ.

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Received: 2017-4-27

Accepted: 2017-7-31

Published Online: 2017-11-10

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