Bounded set
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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 (not to be confused with a 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 Template:Mvar of a 2-dimensional real space R2Script error: No such module "Check for unknown parameters". constrained by two parabolic curves x2 + 1Script error: No such module "Check for unknown parameters". and x2 − 1Script error: No such module "Check for unknown parameters". defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded).
Definition in the real numbers
A set Template:Mvar of real numbers is called bounded from above if there exists some real number Template:Mvar (not necessarily in Template:Mvar) such that k ≥ sScript error: No such module "Check for unknown parameters". for all Template:Mvar in Template:Mvar. The number Template:Mvar is called an upper bound of Template:Mvar. The terms bounded from below and lower bound are similarly defined.
A set Template:Mvar 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 Template:Mvar of a metric space (M, d)Script error: No such module "Check for unknown parameters". is bounded if there exists r > 0Script error: No such module "Check for unknown parameters". such that for all Template:Mvar and Template:Mvar in Template:Mvar, we have d(s, t) < rScript error: No such module "Check for unknown parameters".. The metric space (M, d)Script error: No such module "Check for unknown parameters". is a bounded metric space (or Template:Mvar is a bounded metric) if Template:Mvar is bounded as a subset of itself.
- Total boundedness implies boundedness. For subsets of RnScript error: No such module "Check for unknown parameters". the two are equivalent.
- A metric space is compact if and only if it is complete and totally bounded.
- A subset of Euclidean space RnScript error: No such module "Check for unknown parameters". is compact if and only if it is closed and bounded. This is also called the Heine-Borel theorem.
Boundedness in topological vector spaces
Script error: No such module "Labelled list hatnote". 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 Template:Mvar of a partially ordered set Template:Mvar is called bounded above if there is an element Template:Mvar in Template:Mvar such that k ≥ sScript error: No such module "Check for unknown parameters". for all Template:Mvar in Template:Mvar. The element Template:Mvar is called an upper bound of Template:Mvar. The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.)
A subset Template:Mvar of a partially ordered set Template:Mvar 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 Template:Mvar but also one of the set Template:Mvar as subset of Template:Mvar.
A bounded poset Template:Mvar (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 Template:Mvar of a bounded poset Template:Mvar with as order the restriction of the order on Template:Mvar is not necessarily a bounded poset.
A subset Template:Mvar of RnScript error: No such module "Check for unknown parameters". is bounded with respect to the Euclidean distance if and only if it bounded as subset of RnScript error: No such module "Check for unknown parameters". with the product order. However, Template:Mvar may be bounded as subset of RnScript error: No such module "Check for unknown parameters". 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.
See also
References
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