Editing Bounded set (section)

{{Short description|Collection of mathematical objects of finite size}}{{Noinline|date=November 2023}}[[Image:Bounded unbounded.svg|right|thumb|An [[artist's impression]] of a bounded set (top) and of an unbounded set (bottom). The set at the bottom continues forever towards the right.]]

In [[mathematical analysis]] and related areas of [[mathematics]], a [[Set (mathematics)|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_(mathematics)|metric]].

''[[Boundary (topology)|Boundary]]'' is a distinct concept; for example, a [[circle]] (not to be confused with a [[Disk (mathematics)|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 {{mvar|S}} of a 2-dimensional real space {{math|'''R'''{{sup|2}}}} constrained by two parabolic curves {{math|''x''{{sup|2}} + 1}} and {{math|''x''{{sup|2}} − 1}} defined in a [[Cartesian coordinate system]] is closed by the curves but not bounded (so unbounded).