Return to Revisions

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 pairwise intersections, and assume that they 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.