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Inmathematical analysis and related areas ofmathematics, aset is calledbounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is calledunbounded. The word "bounded" makes no sense in a general topological space without a correspondingmetric.
Boundary is a distinct concept; for example, acircle (not to be confused with adisk) in isolation is a boundaryless bounded set, while thehalf plane is unbounded yet has a boundary.
A bounded set is not necessarily aclosed set and vice versa. For example, a subsetS of a 2-dimensional real spaceR2 constrained by two parabolic curvesx2 + 1 andx2 − 1 defined in aCartesian coordinate system is closed by the curves but not bounded (so unbounded).
A setS ofreal numbers is calledbounded from above if there exists some real numberk (not necessarily inS) such thatk ≥ s for alls inS. The numberk is called anupper bound ofS. The termsbounded from below andlower bound are similarly defined.
A setS isbounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in afinite interval.
AsubsetS of ametric space(M,d) isbounded if there existsr > 0 such that for alls andt inS, we haved(s,t) <r. The metric space(M,d) is abounded metric space (ord is abounded metric) ifM is bounded as a subset of itself.
Intopological vector spaces, a different definition for bounded sets exists which is sometimes calledvon Neumann boundedness. If the topology of the topological vector space is induced by ametric which ishomogeneous, as in the case of a metric induced by thenorm ofnormed vector spaces, then the two definitions coincide.
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. Note that this more general concept of boundedness does not correspond to a notion of "size".
A subsetS of a partially ordered setP is calledbounded above if there is an elementk inP such thatk ≥s for alls inS. The elementk is called anupper bound ofS. The concepts ofbounded below andlower bound are defined similarly. (See alsoupper and lower bounds.)
A subsetS of a partially ordered setP is calledbounded if it has both an upper and a lower bound, or equivalently, if it is contained in aninterval. Note that this is not just a property of the setS but also one of the setS as subset ofP.
Abounded posetP (that is, by itself, not as subset) is one that has a least element and agreatest element. Note that this concept of boundedness has nothing to do with finite size, and that a subsetS of a bounded posetP with as order therestriction of the order onP is not necessarily a bounded poset.
A subsetS ofRn is bounded with respect to theEuclidean distance if and only if it bounded as subset ofRn with theproduct order. However,S may be bounded as subset ofRn with thelexicographical order, but not with respect to the Euclidean distance.
A class ofordinal numbers is said to be unbounded, orcofinal, 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.