Bounded set


In mathematical analysis and related areas of mathematics, a 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.
Boundary is a distinct concept; for example, a circle 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 of a 2-dimensional real space constrained by two parabolic curves and defined in a Cartesian coordinate system is closed by the curves but not bounded.

Definition in the real numbers

A set of real numbers is called bounded from above if there exists some real number such that for all in. The number is called an upper bound of. The terms bounded from below and lower bound are similarly defined.
A set 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

A subset of a metric space is bounded if there exists such that for all and in, we have. The metric space is a bounded metric space if is bounded as a subset of itself.
In topological vector spaces, 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 which is homogeneous, as in the case of a metric induced by the 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 of a partially ordered set is called bounded above if there is an element in such that for all in. The element is called an upper bound of. The concepts of bounded below and lower bound are defined similarly.
A subset of a partially ordered set 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 but also one of the set as subset of.
A bounded poset 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 of a bounded poset with as order the restriction of the order on is not necessarily a bounded poset.
A subset of is bounded with respect to the Euclidean distance if and only if it bounded as subset of with the product order. However, may be bounded as subset of with the lexicographical order, but not with respect to the Euclidean distance.
A class of ordinal numbers 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.