Ring generated by
From Maths
Here ring refers to a ring of sets
Contents
[hide]Definition
Given any class of sets A, there exists a unique ring R_0 such that E\subseteq R_0 and such that if R is any ring with E\subseteq R and R\ne R_0 then R_0\subset R
We call R_0 the ring generated by A, often denoted R(A)
Proof
Since \mathcal{P}(X) (where A is a collection of subsets of X) is a ring (infact an algebra) we know that a ring containing A exists.
Since the intersection of any collection of rings is a ring (see the theorem here), it is clear that the intersection of all rings containing A is the required ring R_0.
Important theorems
Every set in R(A) can be finitely covered by sets in A
[Expand]
If A is any class of sets, then every set in R(A) can be covered by a finite union of sets in A