Closure (mathematics)

In mathematics, the term "closure" refers to the property of a set under a particular operation, meaning that performing that operation on members of the set always produces another member of the same set. If a set exhibits this property with respect to a given operation, we say the set is "closed under" that operation.

More formally, let S be a set and * be a binary operation defined on S. The set S is said to be closed under the operation * if for all elements a and b in S, the result of a * b is also an element of S. This is often written as:

∀ a, b ∈ S : a * b ∈ S

The concept of closure extends beyond binary operations. It can be applied to any operation (unary, ternary, etc.) and to more general algebraic structures. For example, a set can be closed under a unary operation (an operation on a single element).

Closure is a fundamental concept in various branches of mathematics, including set theory, abstract algebra, topology, and analysis. It's critical for defining algebraic structures like groups, rings, and fields, where closure under specific operations (addition, multiplication, etc.) is a defining axiom. The concept of closure also relates to the notion of a "closed set" in topology, where a set contains all its limit points.

If a set is not closed under an operation, applying the operation to elements of the set can produce results that lie outside the set.

The term "closure" can also refer to the smallest set containing a given set that is closed under a particular operation. This smallest set is called the closure of the original set under that operation. For example, the transitive closure of a relation is the smallest transitive relation containing the original relation.