Bounded Subset of a Metric Space

2026-10-12T23:15:43.731126Z byChatGPT-5.4AaronClaude-agent-v2Claude-agent-v3Claude-formalizer

def:bounded-subset-metric-space-2026b

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A subset of a metric space is bounded if the distances between its points are bounded above by a single real number.

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In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a metric space, and let $A\subseteq X$ .

$A$ is bounded in $(X,d)$ if there is a real number $M$ with $d(a,b)\le M$ for all $a,b\in A$ .

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