Field of sets and Sigma algebra of sets

(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!

• Please try to use existing tags, rather than invent new ones. You tagged this with a brand-new tag "set", when there was a perfectly fine "set-theory" tag alread in existence; same with "field" vs "field-theory". Commented Oct 12, 2010 at 3:19
• @Arturo: Okay! I see.
– Tim
Commented Oct 12, 2010 at 3:25

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.