Bounded set

A subset of a metric space is bounded if its points lie within some finite distance of one another. Equivalently, it fits inside a ball of finite radius.

Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Boundedness is defined using the distance function of a metric space.
Finite diameter: A set has finite diameter when the distances between all pairs of its points have a finite upper bound. This is the direct pairwise-distance test for boundedness.
Compact space: A topological space is compact when every open cover has a finite subcover. Compact subsets of metric spaces are bounded, but bounded subsets need not be compact.
Uniform continuity: A function is uniformly continuous if one input-distance threshold works for every point in its domain. On bounded domains, continuity alone need not imply uniform continuity, but compactness does.
Heine–Borel theorem: In Euclidean space, a subset is compact exactly when it is closed and bounded. It makes boundedness part of a precise compactness test in finite-dimensional Euclidean spaces.
Metric: A function assigning distances to pairs of points while satisfying the metric axioms. Its values determine whether a subset has finite diameter.
Ball containment: A set is contained in a ball when every one of its points lies within a fixed radius of a center. In a metric space, boundedness is equivalent to containment in some finite-radius ball.
Totally bounded space: A metric space is totally bounded if finitely many balls of each positive radius cover it. Total boundedness requires increasingly fine finite covers, stronger than mere boundedness.
Arzelà–Ascoli theorem: A theorem characterizing relatively compact families of continuous functions through boundedness and equicontinuity. Uniform boundedness of function values is one condition in its compactness criterion.
Closed interval: A real interval containing both of its finite endpoints. Every closed bounded interval in the real line is compact.

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