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When a set ''S'' is not closed under some operation, one can usually find the smallest subset containing it which is closed. This new closed set is called the closure of ''S''. For example, the closure under subtraction in the reals of the natural numbers is the Integers . Note that the set ''S'' must be a subset of a closed set in order for the closure operator to be defined. It is important here that the reals be closed under subtraction. The two uses of the word "closure" should not be confused. The former usage refers to the property of being closed, and the latter refers to the process of creating a privileged closed set out of one that isn't closed. In short, the closure of a set satisfies the axiom of closure. CLOSED SETS A set is closed under an operation if that operation returns a member of the set when evaluated on members of the set. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known as the axiom of closure. For example, one may define a Group as a set with a binary product such that the product of any two elements of the group is again an element. However the modern definition of an operation makes this axiom superfluous; an N-ary Operator on ''S'' is just a subset of ''S''''n''+1. By its very definition, an operator on a set cannot have values outside the set. Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarily imply closure on all subsets. Thus a subgroup of a group is a subset on which the binary product satisfies the closure axiom. An operation of a different sort is that of adjoining to a subset of a Topological Space the Limit Point s of the subset (if the space is First Countable , it suffices to consider the Limits Of Sequences but in general one must consider limits of Nets ). A set that is closed under this operation is usually just referred to as a Closed Set in the context of Topology . Without any further qualification, the phrase usually means closed in this sense. CLOSURE OPERATOR See Also: closure operator Given an operation on a set ''X'', one can define the closure ''C''(''S'') of a subset ''S'' in ''X'' to be the smallest subset closed under that operation that contains ''S'' as a subset. For example, the closure of a subset of a group is the subgroup Generated by that set. The closure of sets with respect to some operation defines a closure operator on the subsets of ''X''. The closed sets can be determined from the closure operator; a set is closed iff it is equal to its own closure. Typical structural properties of all closure operations are:
An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closure of some object. These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set. EXAMPLES
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