Examples of Seperable and Non Separable Topologies on the Real Line

Letbe the set of real rational numbers.is countable and dense inso the set of real numbers with the usual topology is seperable (takeandinwithThenandand).

Letbe the discrete topology onEvery subset ofwith this topology is both open and closed. The only dense subset ofisitself, butis not countable, hencewith the discrete metric is not seperable.