A space $X$ is second countable if it has a countable basis. We say that a space $X$ is separable if there is a set $Y\subseteq{X}$ such that $Y$ is countable and dense in $X$. Show that if $X$ is a metrizable space then $X$ is separable if and only if $X$ is second countable. Then show that the set of real numbers with arrow topology is a separable space.
I've seen proofs of this on a metric space, but how do I extend it to a general topology? And how do I show that the arrow topology is separable when I know it's not second countable?