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When an Atomic and Complete Algebra of Sets is a Field of Sets with Nowhere Dense Boundary

Abstract We consider pairs 〈A, H(A)〉 where A is an algebra of sets from some class called the class of algebras of type 〈κ, λ〉 and where H(A) is the ideal of hereditary sets of A. We characterize which of the above pairs are topological, that is, which are fields of sets with nowhere dense boundary for some topology together with the ideal of nowhere dense sets for this topology. Making use of a theorem of Fichtenholz and Kantorovich which says that in P(κ) there is an independent family of cardinality 2 κ , we construct an example of a pair 〈algebra, ideal〉 with complete quotient algebra and the hull property but not topological. This countrexample, given in ZFC, provides the complete solution of a problem posed in [Balcerzak, Bartoszewicz, Ciesielski, Real Anal. Exchange 29: 265–273, 2003/04]. Such an algebra was constructed in [Bartoszewicz, Topology Appl. 149: 9–15, 2005] under some aditional set theoretic assumption.