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Part of 1974 Miklós Schweitzer

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Miklos Schweitzer 1974_1

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11/12/2008
Let F \mathcal{F} be a family of subsets of a ground set X X such that ∪F∈FF=X \cup_{F \in \mathcal{F}}F=X, and (a) if A,B∈F A,B \in \mathcal{F}, then A∪B⊆C A \cup B \subseteq C for some C∈F; C \in \mathcal{F}; (b) if An∈F  (n=0,1,...) ,B∈F, A_n \in \mathcal{F} \;(n=0,1,...)\ , B \in \mathcal{F}, and A0⊂A1⊂..., A_0 \subset A_1 \subset..., then, for some k≥0,  An∩B=Ak∩B k \geq 0, \;A_n \cap B=A_k \cap B for all n≥k n \geq k. Show that there exist pairwise disjoint sets Xγ  (γ∈Γ ){ X_{\gamma} \;( \gamma \in \Gamma}\ ), with X=∪{Xγ:  γ∈Γ }, X= \cup \{ X_{\gamma} : \;\gamma \in \Gamma \ \}, such that every Xγ X_{\gamma} is contained in some member of F \mathcal{F}, and every element of F \mathcal{F} is contained in the union of finitely many Xγ X_{\gamma}'s. A. Hajnal
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