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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 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 S of a 2-dimensional real space R2 constrained by two parabolic curves x2 + 1 and x2 - 1 defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded).