Now, if we intersect with anything else in , we get back. Consider the following pairwise intersections
Since intersection is a commutative operation on sets, we have accounted for all possible pairwise intersections of elements of . Notice that every intersection is an element of . Thus, is closed with respect to finite intersection.
By inspection, an arbitrary union of elements in can never yield anything other than the elements of . More rigorously, any union containing must be equal to . Any union containing but not must be equal to . Any union containing just and is equal to . Obviously any union only containing is just . Hence, is closed with respect to arbitrary union.