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Bounded set
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{{Short description|Collection of mathematical objects of finite size}}{{Noinline|date=November 2023}}[[Image:Bounded unbounded.svg|right|thumb|An [[artist's impression]] of a bounded set (top) and of an unbounded set (bottom). The set at the bottom continues forever towards the right.]] In [[mathematical analysis]] and related areas of [[mathematics]], a [[Set (mathematics)|set]] is called '''bounded''' if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called '''unbounded'''. The word "bounded" makes no sense in a general topological space without a corresponding [[Metric_(mathematics)|metric]]. ''[[Boundary (topology)|Boundary]]'' is a distinct concept; for example, a [[circle]] (not to be confused with a [[Disk (mathematics)|disk]]) in isolation is a boundaryless bounded set, while the [[half plane]] is unbounded yet has a boundary. A bounded set is not necessarily a [[closed set]] and vice versa. For example, a subset {{mvar|S}} of a 2-dimensional real space {{math|'''R'''{{sup|2}}}} constrained by two parabolic curves {{math|''x''{{sup|2}} + 1}} and {{math|''x''{{sup|2}} β 1}} defined in a [[Cartesian coordinate system]] is closed by the curves but not bounded (so unbounded). == Definition in the real numbers == [[File:Illustration of supremum.svg|thumb|upright=1.6|A real set with upper bounds and its [[supremum]].]] A set {{mvar|S}} of [[real number]]s is called ''bounded from above'' if there exists some real number {{mvar|k}} (not necessarily in {{mvar|S}}) such that {{math|''k'' β₯ '' s''}} for all {{mvar|s}} in {{mvar|S}}. The number {{mvar|k}} is called an '''upper bound''' of {{mvar|S}}. The terms ''bounded from below'' and '''lower bound''' are similarly defined. A set {{mvar|S}} is '''bounded''' if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a [[interval (mathematics)|finite interval]]. == Definition in a metric space == A [[subset]] {{mvar|S}} of a [[metric space]] {{math|(''M'', ''d'')}} is '''bounded''' if there exists {{math|''r'' > 0}} such that for all {{mvar|s}} and {{mvar|t}} in {{mvar|S}}, we have {{math|''d''(''s'', ''t'') < ''r''}}. The metric space {{math|(''M'', ''d'')}} is a ''bounded'' metric space (or {{mvar|d}} is a ''bounded'' metric) if {{mvar|M}} is bounded as a subset of itself. *[[Total boundedness]] implies boundedness. For subsets of {{math|'''R'''{{sup|''n''}}}} the two are equivalent. *A metric space is [[compact space|compact]] if and only if it is [[Complete metric space|complete]] and totally bounded. *A subset of [[Euclidean space]] {{math|'''R'''{{sup|''n''}}}} is compact if and only if it is [[closed set|closed]] and bounded. This is also called the [[Heine-Borel theorem]]. == Boundedness in topological vector spaces == {{main|Bounded set (topological vector space)}} In [[topological vector space]]s, a different definition for bounded sets exists which is sometimes called [[von Neumann bounded]]ness. If the topology of the topological vector space is induced by a [[metric (mathematics)|metric]] which is [[homogeneous metric|homogeneous]], as in the case of a metric induced by the [[norm (mathematics)|norm]] of [[normed vector spaces]], then the two definitions coincide. ==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'' β₯ ''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. == See also == *[[Bounded domain]] *[[Bounded function]] *[[Local boundedness]] *[[Order theory]] *[[Totally bounded]] ==References== *{{cite book |first1=Robert G. |last1=Bartle |author-link1=Robert G. Bartle |first2=Donald R. |last2=Sherbert |title=Introduction to Real Analysis |location=New York |publisher=John Wiley & Sons |year=1982 |isbn=0-471-05944-7 }} *{{cite book |first=Robert D. |last=Richtmyer |author-link=Robert D. Richtmyer |title=Principles of Advanced Mathematical Physics |publisher=Springer |location=New York |year=1978 |isbn=0-387-08873-3 }} {{Metric spaces}} [[Category:Functional analysis]] [[Category:Mathematical analysis]] [[Category:Order theory]]
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