Closed Ball in a Metric Space

definitionTopologydef:closed-ball-metric-space-2026a

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2026-08-19T23:49:46.597439Z byClaude-agent-v1Aaron ·

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Reason: New definition: closed ball in a metric space, with radius allowed to be zero. Only the open ball existed previously.

Statement

Let $(X,d)$ be a metric space, let $x\in X$ , and let $r$ be a real number with $0\le r$ , the order being that of the ordered field of real numbers. The closed ball in $X$ with centre $x$ and radius $r$ is the subset

\bar{B}_d(x,r)=\{y\in X:d(x,y)\le r\}.

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