Note that while the ordering on " P " [ " D " ] is given by the subset relation, least upper bounds do not in general coincide with unions.
42.
If C is a subset of X that is bounded above and has no least upper bound, let D be the union of all separation on X . This contradicts the connectedness of X.
43.
A partially ordered set with least element "'0 "'is called "'atomistic "'if every element is the least upper bound of a set of atoms.
44.
Then just argue that the least upper bound of the resulting directed set is a least upper bound of the original set .-- talk ) 00 : 24, 30 May 2011 ( UTC)
45.
Then just argue that the least upper bound of the resulting directed set is a least upper bound of the original set .-- talk ) 00 : 24, 30 May 2011 ( UTC)
46.
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.
47.
It is usually taken as an axiom in synthetic constructions of the real numbers ( see least upper bound axiom ), and it is also intimately related to the construction of the real numbers using Dedekind cuts.
48.
:One important property of R is the least upper bound property-- in fact any ordered field with the LUB property is isomorphic to R . That property is quite important for large chunks of analysis.
49.
Naturally, one has a special interest in those domains of computations in which all consistent specifications " converge ", i . e . in orders in which all directed sets have a least upper bound.
50.
From the article on Power domains : " P " [ D ] is the collection of downward-closed subsets of domain D that are also closed under existing least upper bounds of directed sets in D.
