<P> A set is closed under an operation if the 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 operator obeying several axioms, including an axiom 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 operation on S is just a subset of S. By its very definition, an operator on a set cannot have values outside the set . </P> <P> 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 and the unary operation of inversion satisfy the closure axiom . </P> <P> An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space is first - countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at least 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 . Closed intervals like (1, 2) = (x: 1 ≤ x ≤ 2) are closed in this sense . </P> <P> A partially ordered set is downward closed (and also called a lower set) if for every element of the set all smaller elements are also in it; this applies for example for the real intervals (− ∞, p) and (− ∞, p), and for an ordinal number p represented as interval (0, p); every downward closed set of ordinal numbers is itself an ordinal number . </P>

Under which operation is the set of positive rational numbers not closed