# 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 $S$.

$\blacksquare$