GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 1996, 543-546

NON-BAIRE UNIONS IN CATEGORY BASES

J. HEJDUK

Abstract. We consider a partition of a space X consisting of a meager subset of X and obtain a sufficient condition for the existence of a subfamily of this partition which gives a non-Baire subset of X. The condition is formulated in terms of the theory of J. Morgan [1].

All notions concerning category bases come from Morgan’s monograph (see [1]).

We establish the following theorem. Theorem A. Let (X, S) be an arbitrary category base and M(S) be

the σ-ideal of all meager sets in the base (X, S), satisfying the following conditions:

(1) for an arbitrary cardinal number α < card X, the family M(S) is αadditive, i.e., this family is closed under the unions of arbitrary α-sequences of its elements,

(2) there exists a base P of M(S) of cardinality not greater than card X. Thus, if X ∉ M(S), then, for an arbitrary family {X_{t}}_{t∈T} of meager

sets, being a partition of X, there exists a set T′ ⊂ T such that ∪_{t∈T′}X_{t} is not a Baire set.

The proof of this theorem is based on the following lemmas. Lemma 1. If (X, S) is a category base and {A_{α} : α < λ}, where

λ ≤ card S, is the family of essentially disjoint abundant Baire sets, then there exists a family of disjoint regions {B_{α} : α < λ} such that every set A_{α} is abundant everywhere in B_{α} for each α < λ.

The proof of this lemma is similar to that of Theorem 1.5 in [1]. Lemma 2. If (X, S) is a category base, M(S) is the σ-ideal of all meager

sets in the base (X, S) and Φ is a family of subsets of X such that

(1) card Φ > card X, 543 (2) ∀_{Z_{1},Z_{2}∈Φ}(Z_{1} ≠ Z_{2} ⇒ Z_{1} ∩ Z_{2} ∈ M(S)),

1991 Mathematics Subject Classification. Primary 04A03, 03E20, Secondary 28A05, 26A03.

Key words and phrases. Category base, Baire sets, base of an ideal, abundant set, meager set.

1072-947X/96/1100-0543$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

(3) ∀_{Z∈Φ}Z ∉ M(S), then there exists a member of the family Φ which is not a Baire set. Proof. Let us suppose that all members of the family Φ are Baire sets. First of all, we observe that card Φ ≤ card S. Indeed, for each Z ∈ Φ, by the fundamental theorem [1, Ch. 1], we can consider a region V such that Z is abundant everywhere in V. By Theorem 1.3 in [1], we claim that if Z_{1} ≠ Z_{2}, then V_{1} ≠ V_{2}. Hence card Φ ≤ card S. Applying Lemma 1, we obtain a family Φ^{∗} of disjoint nonempty regions such that card Φ^{∗} =card Φ. In that case, we see that card Φ≤card X. This contradicts condition (1).

Lemma 3. If X is an infinite set and Φ_{1} is a family of subsets of X such that

(1) card Φ_{1} ≦ card X,

(2) ∀_{Z∈Φ_{1}}(card Z = card X), then there exists a family Φ_{2} of subsets of X such that

(a) card Φ_{2} > card X,

(b) ∀_{Z_{1},Z_{2}∈Φ_{2}}(Z_{1} ≠ Z_{2} ⇒ card(Z_{1} ∩ Z_{2}) < card X),

(c) ∀_{Y ∈Φ_{1}}∀_{Z∈Φ_{2}}(card(Z ∩ Y) = card X).

This combinatorial lemma is formulated and proved in [2].

Now, we shall prove Theorem A.

We assume that X is not meager and {X_{t}}_{t∈T} is a family of meager sets such that ∪_{t∈T} X_{t} = X. Our goal is to find a set T′ ⊂ T such that ∪_{t∈T′}X_{t} is not a Baire set. We see that card T = card X because, otherwise, X would be meager.

Let P be a base of M(S) of cardinality not greater than card X.

Let Y = {Y ⊂ X : X\Y ∈ P}, K(Y) = {t ∈ T : X_{t} ∩ Y ≠ ∅} and Φ_{1} = (K(Y))_{Y ∈Y}. Then card Φ_{1} ≤ card X. Moreover, we conclude that each member of the family Φ_{1} is of cardinality equal to card X. Indeed, if we have that, for some Y ⊂ X, card K(Y) < card X, then

Y = ∪_{t∈T \K(Y)}(X_{t} ∩ Y) ∪ ∪_{t∈K(Y)}(X_{t} ∩ Y) = ∪_{t∈K(Y)}(X_{t} ∩ Y),

but, by condition (a), we would get that Y is meager. At the same time, X\Y is meager and, finally, X is meager. This contradicts the fact that X is not meager.

By Lemma 3, there exists a family Φ_{2} of subsets of T such that

(a) card Φ_{2} > card T,

(b) ∀_{Z_{1},Z_{2}∈Φ_{2}}(Z_{1} ≠ Z_{2} ⇒ card(Z_{1} ∩ Z_{2}) < card T),

(c) ∀_{Y ∈Φ_{1}}∀_{Z∈Φ_{2}}(card(Z ∩ Y) = card T).

Let X(Z) = ∪_{t∈Z}X_{t} for each Z ∈ Φ_{2}.

We observe that X(Z) is not meager because, otherwise, if X(Z) were meager for some Z ∈ Φ_{2}, then there would exist a member P of base P, such that X(Z) ⊂ P, and then K(X\P) ∩ Z ≠ ∅. Hence there exists t ∈ K(X\P), t ∈ Z. If t ∈ K(X\P), then X_{t}∩(X\P) ≠ ∅, but, simultaneously, X_{t} ⊂ X(Z) ⊂ P. This contradiction ends the proof that X(Z) is not meager. It is clear that, for arbitrary Z_{1}, Z_{2} ∈ Φ_{2}, Z_{1} ≠ Z_{2}, we have X(Z_{1}) ∩ X(Z_{2}) = ∪_{t∈Z_{1}∩Z_{2}}X_{t}.

Since card(Z_{1} ∩ Z_{2}) < card X, therefore X(Z_{1}) ∩ X(Z_{2}) is a meager set. Putting Φ = {X(Z) : Z ∈ Φ_{2}}, we see that

(a) card Φ > card X,

(b) ∀_{X(Z_{1}),X(Z_{2})}(X(Z_{1}) ≠ X(Z_{2}) ⇒ X(Z_{1}) ∩ X(Z_{2}) ∈ M(S)),

(c) ∀_{X(Z)∈Φ}X(Z) ∉ M(S); hence, by Lemma 2, there exists a set Z such that X(Z) is not Baire. This precisely means that there exists T′ ⊂ T such that ∪_{t∈T} X_{t} is not a Baire set.

Now, we need the following well-known auxiliary proposition (see, e.g., [3]).

Lemma 4. Let (X_{i})_{i∈I} be a family of subsets of an infinite set X such that

(a) card(X_{i}) = card X for each i ∈ I,

(b) card(I) ≦ card X. Then there exist sets Y, Z, being a decomposition of X, such that card(Y ∩ X_{i}) = card(Z ∩ X_{i}) = card X for each i ∈ I.

Note that this lemma easily follows from the classical construction of F. Bernstein.

Let (X, S) be a category base.

Definition1. We shall say that a subfamily S′ ⊂ S is a π-base if each region A ∈ S contains a subregion B ∈ S′.

Definition 2 (cf. [4]). We shall say that a set A ⊂ X is not exhausted if, for each meager set P ⊂ X, card(A\P) = card X.

Now, we prove

Theorem B. Let a category base (X, S) satisfy the following conditions:

(a) there exists a π-base S′ such that card S′ ≤ card X and each set of the family S′ is not exhausted,

(b) there exists a base P of the family M(S) of all meager sets of cardinality not greater than card X.

Then the following statements are equivalent:

(a) X is an abundant set,

(b) X contains a non-Baire set. Proof. Only implication (a) ⇒ (b) needs a proof. Let S = {B_{r}}_{r∈R} be a π-base of X such that card R ≤ card X and P = {P_{t}}_{t∈T} a base of M(S) such that card T ≤ card X. Let A = {(B_{r}\P_{t})}_{r∈R,t∈T}. We see that card A ≤ card X and card(B_{r}\P_{t}) = card X for any r ∈ R, t ∈ T. By Lemma 4, there exist sets Y, Z ⊂ X such that X = Y ∪ Z, Y ∩ Z = ∅ and, moreover, Z ∩ (B_{r}\P_{t}) ≠ ∅ ≠ Y ∩ (B_{r}\P_{t}) for any r ∈ R, t ∈ T. It is clear that Y or Z is abundant. Let us assume that the set Y is abundant. We prove that it is not a Baire set. Suppose that Y is a Baire set. By the fundamental theorem, there exists a region V such that Y is abundant everywhere in V. Let V_{r} ⊂ X be a member of a π-base S′ such that V_{r} ⊂ V. It is obvious that Y is abundant everywhere in V_{r}. Since Y is a Baire set, V_{r}\Y is a meager set [1, Th. 1.3]. Thus there exists P_{t} ∈ P such that V_{r}\Y ⊂ P_{t}. Hence we have V_{r}\P_{t} ⊂ Y and (V_{r}\P_{t}) ∩ Z = ∅. This contradicts the fact that the last set should not be empty.

Corollary. In the case where the category base is the family of sets of positive Lebesgue measure over the real line ℝ or the natural topology of this line, we can obtain the existence of a nonmeasurable set or a set without the Baire property.

To conclude, let us recall the following theorem which generalizes one result of Kharazishvili (the proof of this theorem is contained in [5]).

Theorem H. Let (X, S) be a category base on an infinite set X, such that the following conditions are satisfied:

1^{0} M_{0} ⊂ M(S), where M_{0} = {A ⊂ X : card A < card X} and M(S) is a σ-ideal of meager sets,

2^{0} there exists a base of the σ-ideal M(S) of cardinality not greater than card X. Then the following statements are equivalent:

(a) X is a meager set, (b) each subset of X is a Baire set.

For a purely topological version of this theorems see [6].

If we consider Theorems B and H as the criteria of the existence of non-Baire sets in category bases, we discover, by examples 1 and 2 in [4], that they are independent.

References

1. J. Morgan II, Point Set Theory. Marcel Dekker, New York, 1990.

2. A. B. Kharazishvili, Elements of combinatorial theory of infinite sets. (Russian) Tbilisi, 1981.

3. —–, Applications of set theory. (Russian) Tbilisi, 1989.

4. V. M. Rogava, Sets that do not have the Baire property. (Russian) Bull. Acad. Sci. Georgia 142(1991), No. 2, 257–260.

5. J. Hejduk, Non-Baire sets in category bases. Real Anal. Exchange 18(1992/93) No. 2, 448–452.

6. A. B. Kharazishvili, Baire property and its applications. (Russian) Proc. Vekua Inst. Appl. Math., Tbilisi, 1992.

(Received 19.06.1995)

Author’s address:

Institute of Mathematics, Łódź University

ul. Stefana Banacha 22, 90-238 Łódź, Poland