Why is it that for the metric space $(X,d)$ where $X=\Bbb R$ and $d$ is the discrete metric, then $(X,d)$ is not separable? Does this have something to do with $\Bbb R$ being uncountable? Can someone prove why this is true?
2026-04-24 12:28:31.1777033711
Discrete Metric Space is Not Separable
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In the discrete metric all subsets are closed, thus $\operatorname{cl} S=S$. So the only dense subset is $\Bbb R$ itself, which is not countable.