Examples of Seperable and Non Separable Topologies on the Real Line
Let
be the set of real rational numbers.is countable and dense in
so the set of real numbers with the usual topology is seperable (take
and
inwith
Then
and
and
).
Let
be the discrete topology on
Every subset ofwith this topology is both open and closed. The only dense subset ofisitself, butis not countable, hencewith the discrete metric is not seperable.