Completeness axiom

In mathematics the completeness axiom, also called Dedekind completeness of the real numbers, is a fundamental property of the set R of real numbers. It is the property that distinguishes R from other ordered fields, especially from the set of rational numbers.

The axiom states that every non-empty subset S of R that has an upper bound in R has a least upper bound, or supremum, in R. See the article on construction of the real numbers for a full explanation.

The completeness axiom should not be confused with the topological property of completeness of a metric space. The two properties are related, since R, as a metric space with the standard absolute-value metric (where the distance between x and y is |x−y|), does have the latter property as a consequence of its Dedekind completeness. Indeed, R is the completion, in the sense of metric spaces, of the set Q of rational numbers under the absolute-value metric. Thus, the completeness property of metric spaces is one generalization of the completeness axiom itself.

Another generalization focuses on the ordering of the real numbers. In any partially ordered set, the analog of Dedekind completeness is the property that every non-empty subset that is bounded above has a least upper bound; in other words, the same axiom interpreted in greater generality. A partially ordered set with this property is a lattice, specifically a conditionally complete lattice. In practice a stronger property is usually employed: that every subset, whether or not it is empty or bounded above, has a least upper bound. Such a partially ordered set is called a complete lattice.