# Algebra

Let $X$ be a nonempty set. An algebra ${\mathcal {A}}\subseteq 2^{X}$ is a nonempty collection of subsets of $X$ that is closed under finite unions and complements.

By DeMorgan's laws, an algebra is also closed under finite intersections and contains the empty set and $X$ .

## Examples of Algebras

Assume that $X$ is nonempty.

• Given a set $X$ , then $2^{X}$ and $\{\emptyset ,X\}$ are algebras.
• Given a set $X$ , the collection of all finite and cofinite (having finite complement) subsets of $X$ is an algebra.
• A $\sigma$ -algebra is a more restrictive type of algebra. To show an algebra is a $\sigma$ -algebra, it suffices to show closure under countable disjoint unions, which is notably not guaranteed by the definition of an algebra.

## Non-examples

• Given a topological space $(X,\tau )$ , the topology $\tau$ is in general not an algebra, for the complement of an open set $U\in \tau$ may fail to be open. For example, in $X=\mathbb {R}$ with the standard topology, the open interval $(0,1)$ is open, but its complement $(0,1)^{c}=(-\infty ,0]\cup [1,+\infty )$ is not.