Editing Bounded set (section)

==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 {{mvar|S}} of a partially ordered set {{mvar|P}} is called '''bounded above''' if there is an element {{mvar|k}} in {{mvar|P}} such that {{math|''k'' ≥ ''s''}} for all {{mvar|s}} in {{mvar|S}}. The element {{mvar|k}} is called an '''upper bound''' of {{mvar|S}}. The concepts of '''bounded below''' and '''lower bound''' are defined similarly.  (See also [[upper and lower bounds]].)

A subset {{mvar|S}} of a partially ordered set {{mvar|P}} is called '''bounded''' if it has both an upper and a lower bound, or equivalently, if it is contained in an [[Interval (mathematics)#Intervals in order theory|interval]]. Note that this is not just a property of the set {{mvar|S}} but also one of the set {{mvar|S}} as subset of {{mvar|P}}.

A '''bounded poset''' {{mvar|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 {{mvar|S}} of a bounded poset {{mvar|P}} with as order the [[Binary_relation#Restriction|restriction]] of the order on {{mvar|P}} is not necessarily a bounded poset.

A subset {{mvar|S}} of {{math|'''R'''{{sup|''n''}}}} is bounded with respect to the [[Euclidean distance]] if and only if it bounded as subset of {{math|'''R'''{{sup|''n''}}}} with the [[product order]]. However, {{mvar|S}} may be bounded as subset of {{math|'''R'''{{sup|''n''}}}} with the [[lexicographical order]], but not with respect to the Euclidean distance.

A class of [[ordinal number]]s is said to be unbounded, or [[Cofinal (mathematics)|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 a subclass of the class of all ordinal numbers.