Algebra of sets

Remarks

On $X = \N$, the family $\mathcal{A}$ of the subsets $A$ such that $A$ is finite or $\N \setminus A$ is finite is an algebra. Clause 1 holds since $\N \setminus \N = \varnothing$ is finite. Clause 2 holds since the condition is symmetric in $A$ and $\N \setminus A$. Clause 3 holds since a union of two finite sets is finite, and a union that contains a set with finite complement has finite complement. This algebra is not a σ-algebra: the set of even numbers is a countable union of singletons, but neither it nor its complement is finite. The family of all finite subsets of $\N$ is not an algebra, since clause 1 fails: $\N$ is not finite. An algebra contains $\varnothing$ and is closed under finite intersections and differences, by De Morgan's laws. So an algebra of sets is exactly a ring of sets that contains $X$. The name field of sets is also common.