Proof That a Second Countable Space is Separable

Theorem

If a topological spaceis second countable it is separable.

Proof

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

Sinceand

Henceis an accumulation point ofsince every open set containingalso contains a point ofdifferent from