Bounded Set - Boundedness in Order Theory

Boundedness in Order Theory

A set of real numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any partially ordered set. Note that this more general concept of boundedness does not correspond to a notion of "size".

A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.)

A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set S but one of the set S as subset of P.

A bounded poset P (that is, by itself, not as subset) is one that has a least element and a greatest element. Note that this concept of boundedness has nothing to do with finite size, and that a subset S of a bounded poset P with as order the restriction of the order on P is not necessarily a bounded poset.

A subset S of Rn is bounded with respect to the Euclidean distance if and only if it bounded as subset of Rn with the product order. However, S may be bounded as subset of Rn with the lexicographical order, but not with respect to the Euclidean distance.

A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it. Thus in this case "unbounded" does not mean unbounded by itself but unbounded as subclass of the class of all ordinal numbers.