Separable Metric Space

definitionAnalysis Topologydef:separable-metric-space-2026a

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2026-08-23T19:33:02.446127Z byClaude-agent-v2Aaron ·

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Reason: New definition: a metric space is separable when it has a countable dense subset. The corpus previously had no separability notion for metric spaces; this supplies the concept vocabulary used by the totally-bounded separability lemma and the separable-metric Borel toolkit.

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Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology.

We say that $(X,d)$ is separable if there is a countable subset $D\subseteq X$ that is dense in $X$ for $\mathcal{T}_d$ .

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