Theorem 1   [1] Every second countable space is separable.

Proof. Let $X$ be a second countable space and let $\cal B$ be a countable base. For every non-empty set $B$ in $\cal B$ , choose a point $x_B\in B$ . The set $A$ of all such points $x_B$ is clearly countable and it's also dense since any open set intersects it and thus the whole space is the closure of $A$ . That is, $A$ is a countably dense subset of $X$ . Therefore, $X$ is separable.