Editing Bounded set

{{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}} − 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'' ≥ '' 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'' ≥ ''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]]