Closure (mathematics) - Closed Sets

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 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 Sn+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 and the unary operation of inversion satisfy the closure axiom.

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 = {x : 1 ≤ x ≤ 2} are closed in this sense.

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.

Upward closed and upper set are defined similarly.