Open Set Characterization of Denseness/Open Ball
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Theorem
Let $\struct {X, d}$ be a metric space.
Let $\tau_d$ be the topology induced by the metric $d$.
Let $S \subseteq X$.
Then $S$ is (everywhere) dense in $\struct {X, \tau_d}$ if and only if every open ball contains an element of $S$.
Proof
By Open Balls form Basis for Open Sets of Metric Space, the set of open balls are an analytic basis for the topology $\tau_d$.
By Analytic Basis Characterization of Denseness then:
- $S$ is (everywhere) dense in $\struct {X, \tau_d}$ if and only if every open ball contains an element of $S$.
$\blacksquare$