Examples
If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.
If (X,≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) is closed under both intersections and unions. However, in general it will not be closed under differences of sets.
The open sets and closed sets of any topological space are closed under both unions and intersections.
If T is any transformation of a space, then the sets that are mapped into themselves by T are closed under both unions and intersections.
If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form a ring of sets.
Read more about this topic: Ring Of Sets
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—Alexander Pope (1688–1744)
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