Closure (mathematics)

In mathematics, closure describes the case when the results of a mathematical operation are always defined. For example, in ordinary arithmetic, addition on real numbers has closure: whenever one adds two numbers, the answer is a number. The same is true of multiplication. Division does not have closure, because division by 0 is not defined. In the natural numbers, subtraction does not have closure, but in the integers, subtraction does have closure. Subtraction of two numbers can produce a negative number, which is not a natural number, but is an integer.

One can sometimes make closure of a mathematical object by including new elements to it. The integers are a closure of the natural numbers by including negative numbers. The real numbers are a closure of the rational numbers by including square roots of positive numbers. The complex numbers are a closure of the real numbers by including square roots of negative numbers. Since every non-zero polynomial has a root in the complex numbers, the complex numbers are also the algebraic closure of the real numbers. One can express this relationship as ${\displaystyle {\overline {\mathbb {R} }}=\mathbb {C} }$.^[1]

Sometimes, if one includes an element to make closure, it makes more changes. For example, if one includes infinity, ${\displaystyle \infty =1/0}$ (that is, closure of division), the laws of addition and subtraction are changed. There is no inversion of addition for ${\displaystyle \infty }$.

Ordinary closure is called finite closure. There is also infinite closure. The definition of a topological space mentions infinite closure. Open spaces have (finite) closure of intersection. Open sets have infinite closure of union. That is, in mathematical notation, if A₀, and A₁ and ... A_n ... are open sets, then B and C are open sets:

• ${\displaystyle B=A_{0}\cap A_{1}}$

• ${\displaystyle C={\cup _{n}}A_{n}}$

In point-set topology, given a set S, the set containing all points of S along with its limit points is called the topological closure of S. This is sometimes written as ${\displaystyle {\overline {S}}}$.^[1] The closure of S is also the smallest closed set containing S.^[2][3]

• Group (mathematics)

1. ↑ ^{1.0 1.1} "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-12.
2. ↑ Weisstein, Eric W. "Set Closure". mathworld.wolfram.com. Retrieved 2020-10-12.
3. ↑ Weisstein, Eric W. "Closure". mathworld.wolfram.com. Retrieved 2020-10-12.