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# 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. 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 $\mathcal{R}$.

## Mathematics Subject Classification

03E20*no label found*28A05

*no label found*

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