(1). According to Wikipedia, a field of subsets of $X$ is defined to be a non-empty subset of the power set of $X$ closed under the intersection and union of pairs of sets and under complements of individual sets. I was wondering if there is any redundancy in this definition?
Can it be defined to be closed only under pairwise intersection and pairwise union? Or only closed under complement and pairwise intersection (or pairwise union)? If yes in each case, how can we show it also closed under the other operation?
(2). A sigma algebra of subsets of $X$ is defined to be a nonempty subset of the power set of $X$ closed under complementation and under countable union.
Can it be defined to be closed under countable union and countable intersection? If yes, how can we show it is subsequently closed under complement?
Thanks and regards!
Best Answer
No, you can't get complements from unions and intersections. For example, let $X$ be a nonempty set. Then $\{\emptyset\}$ is nonempty, closed under (arbitrary) intersection and union, but not closed under complements.
You can get intersections from unions and complements using De Morgan's laws. To get an intersection, just take the complement of the union of the complements. Similarly, you get unions from intersections and complements. So yes, the definition of field of subsets was redundant.