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In a metric space, every compact set is bounded
Compactness implies boundedness: Let be a metric space and let be a compact set. Then is a bounded set; equivalently, , where is the diameter induced by .
This is a first step toward “compact sets behave like finite sets” in metric settings, and it complements compactness implies total boundedness.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A bounded set in a metric space is a subset such that either , or there exist and such that
where is the open ball of radius around .
The diameter of a subset of a metric space is
where denotes the supremum (and the value may be ). By convention, .
Compactness implies total boundedness: Let be a metric space and let be compact. Then is totally bounded: for every there exists a finite set such that
where denotes the open ball.