examples of ring of sets

Every field of sets is a ring of sets. Below are some examples of rings of sets that are not fields of sets.

1. Let $A$ be a non-empty set containing an element $a$ . Let $\mathcal{R}$ be the family of subsets of $A$ containing $a$ . Then $\mathcal{R}$ is a ring of sets, but not a field of sets, since $\lbrace a\rbrace \in \mathcal{R}$ , but $A-\lbrace a\rbrace \notin \mathcal{R}$ .
2. The collection of all open sets of a topological space is a ring of sets, which is in general not a field of sets, unless every open set is also closed. Likewise, the collection of all closed sets of a topological space is also a ring of sets.
3. A simple example of a ring of sets is the subset $\{ \{a\}, \{a,b\} \}$ of $2^{\{a,b\}}$ . That this is a ring of sets follows from the observations that $\{a\} \cap \{a,b\} = \{a\}$ and $\{a\} \cup \{a,b\} = \{a,b\}$ . Note that it is not a field of sets because the complement of $\{a\}$ , which is $\{b\}$ , does not belong to the ring.
4. Another example involves an infinite set. Let $A$ be an infinite set. Let $\mathcal{R}$ be the collection of finite subsets of $A$ . Since the union and the intersection of two finite set are finite sets, $\mathcal{R}$ is a ring of sets. However, it is not a field of sets, because the complement of a finite subset of $A$ is infinite, and thus not a member of $\mathcal{R}$ .