AMS JOURNAL STYLE Volume 000, 0000

THE COVERING NUMBER AND THE TRANSITIVE COVERING NUMBER MAY BE TOTALLY DIFFERENT

JAN KRASZEWSKI

Abstract. We construct a translation invariant σ −ideal T (κ) (where κ is an infinite cardinal number) such that cov_{t}(T (κ)) = 2^{κ} while cov(T (κ)) = cof(T (κ)) = ω_{1}. The constructions can be carried out in ℝ as well.

0. Introduction

In 1938 Rothberger in [2] proved that there is a family of meagre subsets of the real line of size less or equal than the least cardinality of a Lebesgue nonmeasurable set such that its sum is the whole real line (and the same when we replace meagre sets by Lebesgue null sets and a Lebesgue nonmeasurable set by a set without the Baire property). In other words, he showed that cov(Meagre) ≤ non(Null) and cov(Null) ≤ non(Meagre), where cov and non stand for a covering number and a uniformity of a given ideal. As a matter of fact, Rothberger proved more.

Theorem 0.1. Let J and I be translation invariant ideals of subsets of a group G, orthogonal to each other (that is there exist A ∈ J and B ∈ I such that A∪B = G). Then

cov_{t}(J) ≤ non(I).

where non(I) is the minimal cardinality of the subset of G that does not belong to I. □

In this theorem cov_{t}(J) denotes a transitive covering number of an ideal J. The natural question to ask is what in general is a possible difference between a covering number and a transitive covering number of a given ideal. In this paper we show that these two cardinal invariants may be totally different.

1. Definitions and basic properties

We use standard set-theoretical notation and terminology from [1]. In particular, the cardinality of the set of all real numbers is denoted by 𝔠. The cardinality of a set X is denoted by |X|. A power set of a set X is denoted by P(X). If κ is a cardinal number then cf(κ) denotes its cofinality.

Let (G, +) be an infinite abelian group. We consider a σ −ideal J of subsets of G which is proper and contains all singletons. Moreover, we assume that J is

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page

2000 Mathematics Subject Classification. 03E05, 20K99.

Key words and phrases. Abelian groups, ideals, covering number, Hamel basis.

translation invariant (i.e. (∀A ∈ J)(∀g ∈ G) A + g = {a + g : a ∈ A} ∈ J) and symmetric (i.e. (∀A ∈ J) − A = {−a : a ∈ A} ∈ J).

We say that a family B ⊆ J is cofinal with J if for each A ∈ J there exists such B ∈ B that A ⊆ B. We also call such a family B a base of J.

For an ideal J we consider the following cardinal numbers

cov(J) = min{|A| : A ⊆ J & ⋃ A = G},
cov_{t}(J) = min{|T | : T ⊆ G & (∃A ∈ J) A + T = G},
cof(J) = min{|B| : B ⊆ J & B is a base of J }.

They are called the covering number, the transitive covering number and the cofinality of J, respectively. Note that the following relations hold:

cov(J) ≤ cov_{t}(J) and cov(J) ≤ cof(J).

For more information about cardinal invariants of ideals on abelian groups and relations between them – see [1].

A set H ⊆ ℝ is called a Hamel basis if it is a basis of (ℝ, +) treated as a linear space over a field ℚ of rational numbers.

2. Cofinality versus transitive covering

In this section we show that transitive covering of an ideal may be totally different from its cofinality.

Theorem 2.1. Let λ be a cardinal number of uncountable cofinality and let ⟨G_{α} : α < λ⟩ be a strictly increasing sequence of subgroups of a group G such that G = ⋃_{α<λ} G_{α}. If J is a σ−ideal of subsets of G generated by the family {G_{α} : α < λ} then cof(J) = cf(λ) and

cov_{t}(J) = inf{|^{G}/_{G_{α}}| : α < λ}.

Proof. Straight from the fact, that the sequence ⟨G_{α} : α < λ⟩ is increasing and cf(λ) is uncountable we can deduce that

J = {A ⊆ G : (∃ξ < λ) A ⊆ G_{ξ}}.

It is a simple observation that J is a translation invariant, symmetric σ−ideal containing singletons. It is also proper because of strict monotonicity of the sequence ⟨G_{α} : α < λ⟩.

Let us fix a given sequence of ordinal numbers ⟨ξ_{α} : α < cf(λ)⟩, cofinal in λ. Then the family {G_{ξ_{α}} : α < cf(λ)} is a base for J. Moreover, no family of elements of J of cardinality strictly less than cf(λ) can be a base of J as all elements of such a family are contained in G_{ξ} for some ξ < λ. Hence cof(J) = cf(λ).

Let us observe that a sequence of cardinal numbers ⟨|^{G}/_{G_{α}}| : α < λ⟩ is decreasing so there exists ζ < λ such that |^{G}/_{G_{α}}| = |^{G}/_{G_{ζ}}| for α ≥ ζ. Let us consider now a set T ⊆ G such that |T | = cov_{t}(J) and there exists A ∈ J such that A + T = G.

THE COVERING NUMBER AND THE TRANSITIVE COVERING NUMBER MAY BE TOTALLY DIFFERENT3

Without loss of generality we may assume that A = G_{ξ} for some ζ ≤ ξ < λ. Then we may get T′ ⊆ T such that (∀t ∈ T′) T′ ∩ (G_{ξ} + t) = {t} and G_{ξ} + T′ = G, that is, T′ is a selector of the cosets. Thus cov_{t}(J) = |T′| = |^{G}/_{G_{ξ}}| and, consequently,

cov_{t}(J) = |^{G}/_{G_{ξ}}| = |^{G}/_{G_{ζ}}| = inf{|^{G}/_{G_{α}}| : α < λ},

which ends the proof. □

As an application of Theorem 2.1 we construct a σ−ideal, the transitive covering of which is in general radically bigger than its cofinality and, consequently, its covering number as well. First, we introduce some necessary notation.

From now on let us fix a Hamel basis H and its enumeration H = {h_{α} : α < 𝔠}. Then every real number x has the unique representation in this basis, i.e.

(∀x ∈ ℝ)(∃! r_{x} ∈ ℚ^{𝔠})(| supp(r_{x})| < ω & x = ∑_{α<𝔠} r_{x}(α)h_{α}),

where supp(r_{x}) = {α : r_{x}(α) ≠ 0}. In order to simplify the notation we replace supp(r_{x}) by supp(x).

Definition. Let {P_{ξ} : ξ < ω_{1}} be a fixed partition of 𝔠 into parts of cardinality 𝔠. Let A be any set. We say that a function f ∈ ℝ^{A} is Hamel-bounded if

(∃ξ < ω_{1})(∀a ∈ A)(supp(f(a)) ⊆ ⋃_{β<ξ} P_{β}).

Then we put HB(A) = {f ∈ ℝ^{A} : f is Hamel − bounded}. One can check that HB(A) is a subgroup of ℝ^{A} with the standard addition of functions. For any function f ∈ HB(A) its Hamel-bound hb (f) is defined as follows:

hb (f) = min{ξ < ω_{1} : (∀a ∈ A)(supp(f(a)) ⊆ ⋃_{β<ξ} P_{β})}.

Let κ be an infinite cardinal number. Let B_{ξ} = {f ∈ HB(κ) : hb (f) ≤ ξ}. Of course, ⟨B_{ξ} : ξ < ω_{1}⟩ is a strictly increasing sequence of subgroups of the group HB(κ) and HB(κ) = ⋃_{ξ<ω_{1}} B_{ξ}. We define T (κ) as a σ −ideal generated by the family {B_{ξ} : ξ < ω_{1}}.

Lemma 2.2. |^{HB(κ)}/_{B_{ξ}} | = 2^{κ} for every ξ < ω_{1}.

Proof. Let us fix B_{ξ} for some ξ < ω_{1}. We consider a set T ⊆ HB(κ) such that (∀t ∈ T) T ∩ (B_{ξ} + t) = {t} and B_{ξ} + T = HB(κ).

Let us fix P ⊆ κ and a real number x such that x ∈ H \ {h_{α} : α ∈ ⋃_{β<ξ} P_{β}}. We define a function f_{P} ∈ HB(κ) as follows:

f_{P} (α) = χ_{P} (α) · x

where χ_{P} denotes the characteristic function of a set P. Then there exists t_{P} ∈ T and g ∈ B_{ξ} such that f_{P} = g + t_{P}. In particular, for each α ∈ P we have

x = f_{P} (α) = g(α) + t_{P} (α).

But we know from the assumption that supp(x) ⊈ ⋃_{β<ξ} P_{β}, so we have supp(x) ⊆ supp(t_{P} (α)) for each α ∈ P. On the other hand, if α ∉ P then f_{P} (α) = 0 and, consequently, supp(t_{P} (α)) = supp(g(α)) ⊆ ⋃_{β<ξ} P_{β} for such α’s.

Let P_{1} and P_{2} be two different subsets of κ and α ∈ P_{1} △ P_{2}. Suppose that t_{P_{1}} = t_{P_{2}} = t. Then

supp(x) ⊆ supp(t(α)) ⊆ ⋃_{β<ξ} P_{β},

which is a contradiction. Hence t_{P_{1}} ≠ t_{P_{2}} and, consequently,

|^{HB(κ)}/_{B_{ξ}} | = |T | ≥ |P(κ)| = 2^{κ},

which ends the proof, as |HB(κ)| = 2^{κ}. □

Corollary 2.3. For every infinite cardinal number κ we have cof(T (κ)) = ω_{1} and cov_{t}(T (κ)) = 2^{κ}.

Proof. It is enough to apply Theorem 2.1 for λ = ω_{1}, G = HB(κ), G_{ξ} = B_{ξ} and J = T (κ). Thanks to Lemma 2.2 we get the result. □

Acknowledgements. The author would like to thank Professor Cichoń for many fruitful discussions which helped him improve the proof.

References

[1]. T. Bartoszyński, H. Judah, Set Theory: On the structure of the real line, A. K. Peters,

Wellesley, Massachusetts (1995). [2]. F. Rothberger, Eine Aquivalenz zwischen der Kontinuumhypothese under der Existenz der

Lusinschen und Sierpinskischen Mengen, Fund. Math. 30 (1938), 215–217.

Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-156 Wrocław, Poland

   E-mail address: kraszew@math.uni.wroc.pl