Show that $X$ is separable.

Show that $X$ is separable.

I'm working through Kunen's Set Theory and I'm not sure how to proceed on
part of one exercise. Let $X$ be compact Hausdorff and $\mathbb{O}_X$ be
the poset of nonempty open sets of $X$ ordered by inclusion. I want to
show that all of
$X$ is separable.
$\mathbb{O}_X$ is $\sigma$-centered.
$\mathbb{O}_X$ is a countable union of filters.
are equivalent. I have 3 $\Rightarrow$ 2 and 1 $\Rightarrow$ 3. For 2
$\Rightarrow$ 1, we have that $\mathbb{O}_X=\bigcup_{n\in\Bbb N} C_n$
where each $C_n$ is centered, but I'm not sure how from these I should
pick the points in the countable dense subset. Also, I haven't used the
compact Hausdorff hypotheses yet and I am not sure how they will come into
play.