
 Dedekind cut  Freepedia (Site not responding. Last check: 20071107) 
  In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is closed upwards. 
  In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it does have the leastupperbound property, i.e., its every nonempty subset that has an upper bound has a least upper bound. 
  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 leastupperbound property. 
 en.freepedia.org /Dedekind_completion.html (506 words) 
