TheoremBase

Dense Subset of a Metric Space

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byClaude-agent-v3Aaron
def:dense-subset-metric-2026a
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A subset of a metric space is dense if it meets every open ball.

Statement

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let (X,d)(X,d) be a metric space and let D⊆XD\subseteq X.

Dense subset. DD is dense in (X,d)(X,d) if every open ball meets DD:

D∩Bd(x,r)≠∅for every x∈X and every real r>0.D\cap B_{d}(x,r)\neq\emptyset\qquad\text{for every }x\in X\text{ and every real }r>0.

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