# Algebra

Let ${\displaystyle X}$ be a nonempty set. An algebra[1] ${\displaystyle {\mathcal {A}}\subseteq 2^{X}}$ is a nonempty collection of subsets of ${\displaystyle X}$ 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 ${\displaystyle X}$.

## Examples of Algebras

Assume that ${\displaystyle X}$ is nonempty.

• Given a set ${\displaystyle X}$, then ${\displaystyle 2^{X}}$ and ${\displaystyle \{\emptyset ,X\}}$ are algebras.
• Given a set ${\displaystyle X}$, the collection of all finite and cofinite (having finite complement) subsets of ${\displaystyle X}$ is an algebra.
• A ${\displaystyle \sigma }$-algebra is a more restrictive type of algebra. To show an algebra is a ${\displaystyle \sigma }$-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 ${\displaystyle (X,\tau )}$, the topology ${\displaystyle \tau }$ is in general not an algebra, for the complement of an open set ${\displaystyle U\in \tau }$ may fail to be open. For example, in ${\displaystyle X=\mathbb {R} }$ with the standard topology, the open interval ${\displaystyle (0,1)}$ is open, but its complement ${\displaystyle (0,1)^{c}=(-\infty ,0]\cup [1,+\infty )}$ is not.

## References

1. Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.