On some algebra of sets in Steprāns strong-Q-sequence model

Let H(A) stand for the ideal of sets that hereditarily belong to a given algebra A of sets. Let S,S0 denote the operations of Marczewski. We say that 〈A,H(A)〉 is inner MB-representable (is topological) if 〈A,H(A)〉=〈S(F),S0(F)〉 for a nonempty F⊂A (for F equal to τ∖{∅} where τ is a topology on ⋃A). We show that the existence of an algebra A isomorphic to P(ω), with 〈A,H(A)〉 inner MB-representable but not topological, and with complete Boolean algebra A/H(A), is independent of ZFC.