:''"Bounded" and "boundary" are distinct concepts; for the latter see

^{''n''} the two are equivalent.
*A metric space is ^{''n''} is compact if and only if it is closed and bounded.

^{''n''} is bounded with respect to the ^{''n''} with the ^{''n''} with the

boundary (topology)
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric obj ...

. A circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...

and related areas of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a set is called bounded if it is, in a certain sense, of finite size. 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
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

.
Definition in the real numbers

A set ''S'' ofreal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s is called ''bounded from above'' if there exists some real number ''k'' (not necessarily in ''S'') such that ''k'' ≥ '' s'' for all ''s'' in ''S''. The number ''k'' is called an upper bound of ''S''. The terms ''bounded from below'' and lower bound are similarly defined.
A set ''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 finite interval.
Definition in a metric space

Asubset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

''S'' of a metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

(''M'', ''d'') is bounded if there exists ''r'' > 0 such that for all ''s'' and ''t'' in ''S'', we have d(''s'', ''t'') < ''r''. (''M'', ''d'') is a ''bounded'' metric space (or ''d'' is a ''bounded'' metric) if ''M'' is bounded as a subset of itself.
* Total boundedness implies boundedness. For subsets of Rcompact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

if and only if it is complete and totally bounded.
*A subset of Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

RBoundedness in topological vector spaces

Intopological vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

which is homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences
Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about th ...

, as in the case of a metric induced by the norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy)
Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...

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 anypartially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

. 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
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

.)
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 also 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
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, t ...

. 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 REuclidean distance
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

if and only if it bounded as subset of Rproduct order
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. However, ''S'' may be bounded as subset of Rlexicographical order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, but not with respect to the Euclidean distance.
A class of ordinal number
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...

s 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 a subclass of the class of all ordinal numbers.
See also

*Bounded function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

* Local boundedness
*Order theory
Order theory is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...

*Totally boundedIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...

References

* *{{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 Mathematical analysis Functional analysis Order theory