# Closure

Closure is a property of an abstract algebraic structure, such as a set, group, ring, or field

## Definition

An algebraic structure $\mathbb{S}$ is said to have closure in a binary operation $\times$ if for any $a,b\in \mathbb{S}$, $a\times b\in \mathbb{S}$. In words, when any two members of $\mathbb{S}$ are combined using the operation, the result also is a member of $\mathbb{S}$.

## Examples

• The real number system $\mathbb{R}$ has closure in addition, subtraction, multiplication, division, exponentiation, and also higher level operations such as $a \uparrow \uparrow b$.
• The rational number system $\mathbb{Q}$ has closure in addition, subtraction, multiplication, and division
• The natural and whole number systems $\mathbb{N}^+,\mathbb{N}^0$ have closure in addition and multiplication.
• The complex number system $\mathbb{C}$ has closure in addition, subtraction, multiplication, division, exponentiation, and also higher level operations such as $a \uparrow \uparrow b$.
• The integral number system $\mathbb{Z}$ has closure in addition, subtraction, multiplication, and exponentiation.