"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology)

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

. A circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.

In mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 and related areas of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 metric.

Definition

A set S of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s is called bounded from above if there is a real number k 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

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

.

Metric space

A subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 S of a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

 (M, d) is bounded if it is contained in a ball

Ball (mathematics)

In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....

 of finite radius, i.e. if there exists x in M and r > 0 such that for all s in S, we have d(x, s) < r. M 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 Rⁿ the two are equivalent.
• A metric space is compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   if and only if it is complete and totally bounded.
• A subset of Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
   Rⁿ is compact if and only if it is closed
  Closed set
  In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
  
   and bounded.

Boundedness in topological vector spaces

In topological vector space

Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

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 (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

 which is homogenous, as in the case of a metric induced by the norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

 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

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

. 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

Greatest element

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...

. 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

Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

 if and only if it bounded as subset of Rⁿ with the product order

Product order

In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product A × B. Giventwo pairs and in A × B, one sets ≤...

. However, S may be bounded as subset of Rⁿ with the lexicographical order

Lexicographical order

In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...

, but not with respect to the Euclidean distance.

A class of ordinal number

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...

s is said to be unbounded, or cofinal

Cofinal (mathematics)

In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition:This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Also, the notion of 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.