Open Set Characterization of Denseness/Open Ball

From ProofWiki
Jump to navigation Jump to search

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 $S$.

$\blacksquare$