A Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that whenever a is in A and x ≤ a, then x is in A as well), B is closed upwards. If a is a member of S then the set { { x in S : x ≤ a }, { x in S : x > a } } is a Dedekind cut that gets identified with a, so that the linearly ordered set S may be regarded as embedded within the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts is strictly bigger than S. Regard one Dedekind cut { A, B } as less than another Dedekind cut { C, D } if A is a proper subset of C, or, equivalently D is a proper subset of B. In this way, the set of all Dedkind cuts is itself a linearly ordered set, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound. Embedding S within a larger linearly ordered set that does have the least-upper-bound property is the purpose.
The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers. A typical Dedekind cut of the rational numbers is given by A = { a in Q : a2 < 2 or a ≤ 0 }, B = { b in Q : b2 ≥ 2 & b > 0 }. This cut represents the real number √ 2 in Dedekind's construction.
More generally, in a partially ordered set S, the set of all nonempty "downwardly closed" subsets (also called "order ideals") is a set partially ordered by inclusion, and in the same way we embed S within a larger partially ordered set that, generally unlike the original set S, does have the least-upper-bound property. This larger poset is called the Dedekind completion of S.
A construction very similar to Dedekind cuts is used for the construction of surreal numbers.
See also:
Generalization: Dedekind completions in posets
Another generalization: surreal numbers
