A has a maximum, nor B has a minimum, the cut is called a gap. A linearly ordered set endowed with the order topology is compact if and only if it has no gap. Surreal numbers A construction resembling Dedekind cuts is used for (one among many possible) constructions of surreal numbers. The relevant notion in this case is a Cuesta-Dutari cut, named after the Spanish mathematician . Partially ordered sets More generally, if S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding of S into L. The notion of complete lattice generalizes the least-upper-bound property of the reals. One completion of S is the set of its downwardly closed subsets, ordered by inclusion. A related completion that preserves all existing sups and infs of S is obtained by the following construction: For each subset A of S, let Au denote the set of upper bounds of A, and let Al denote the set of lower bounds of A. (These operators form a Galois connection.) Then the Dedekind–MacNeille completion of S consists of all subsets A for which (Au)l = A; it is ordered by inclusion. The Dedekind-MacNeille completion is the smallest complete lattice with S embedded in it. Notes References Dedekind, Richard, Essays on the Theory of Numbers, "Continuity and Irrational Numbers," Dover Publications: New York, . Also available at Project Gutenberg. External links Order theory Rational numbers Real numbers