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You seem just to be generating the Boolean subalgebra of the power set generated by your family. If you start with one nonempty non-everything set, you will get 4 sets: the set, its complement, the empty set and the whole space. In general, if you start with a finite collection of sets, then form all possible intersections, to generate the smallest possible nonempty sets, which are pairwise-disjoint. Throw in the complement of the union of them, and you have in effect a collection of n atoms. You can now generate 2ⁿ sets by taking the union of any of these atoms, and this collection is closed under union, intersection and complement, and includes the original sets. So a finite set will generate a finite set this way. This problem has nothing to do with the reals and works with subsets of any set. You have a collection of subsets of a given set X. The power set P(X) is a Boolean algebra with the operations of union, intersection and complement. Your question is asking about the finitely generated subalgebras of P(X). But every finitely genereated Boolean algebra is isomorphic to a finite power set (of the atoms), and has size 2ⁿ, where n is the number of atoms.