Dedekind cut
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