Proof That a Second Countable Space is Separable
If a topological spaceis second countable it is separable.
Supposeis a second countable space. Then there exists a countable baseofFor eachchoosesuch that
The setis countable. Letand letbe an open set containingAt least one setexists such that
Henceis an accumulation point ofsince every open set containingalso contains a point ofdifferent from