Non-baire Unions in Category Bases

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 6∈ M(S), then, for an arbitrary family {Xt}t∈T of meager sets, being a partition of X, there exists a set T ′ ⊂ T such that ∪t∈T ′Xt 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 {Aα : α < λ}, where λ ≤ cardS, is the family of essentially disjoint abundant Baire sets, then there exists a family of disjoint regions {Bα : α < λ} such that every set Aα is abundant everywhere in Bα 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, 1991 Mathematics Subject Classification. Primary 04A03, 03E20, Secondary 28A05, 26A03.