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 formula_1.
Sometimes, if one includes an element to make closure, it makes more changes. For example, if one includes infinity, formula_2 (that is, closure of division), the laws of addition and subtraction are changed. There is no inversion of addition for formula_3.
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:
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 formula_6. The closure of "S" is also the smallest closed set containing "S".